Device, method, and program recording medium for control of air-fuel ratio of internal combustion engine

ABSTRACT

An apparatus for controlling the air-fuel ratio of an internal combustion engine to compensate for the effect of the dead times of an exhaust system including a catalytic converter, etc. and to increase the purifying capability of the catalytic converter. An exhaust-side control unit  7   a  sequentially variably sets a dead time of an exhaust system E depending on the flow rate of an exhaust gas supplied to a catalytic converter  3  and a dead time of an air-fuel ratio manipulating system comprising an internal combustion engine  1  and an engine-side control unit  7   b , and sequentially estimates an output of an O 2  sensor  6  after a total set dead time which is the sum of the above set dead times. The exhaust-side control unit  7   a  sequentially generates a target air-fuel ratio KCMD to converge the output of the O 2  sensor  6  to a target value using the estimated value, and manipulates the air-fuel ratio of the internal combustion engine  1 . Using the set dead time of the exhaust system E as a dead time of an exhaust system model which serves as a basis for estimating the output of the O 2  sensor  6 , a parameter of the exhaust system model is sequentially identified.

TECHNICAL FIELD

The present invention relates to an apparatus for and a method ofcontrolling the air-fuel ratio of an internal combustion engine, and arecording medium storing a program for controlling the air-fuel ratio ofan internal combustion engine.

BACKGROUND ART

There have already been proposed by the applicant of the presentapplication techniques for controlling the air-fuel ratio of an air-fuelmixture to be combusted by an internal combustion engine for convergingthe output of an exhaust gas sensor, e.g., an O₂ sensor (oxygenconcentration sensor), disposed downstream of a catalytic converter, toa predetermined target value (constant value) in order to achieve theappropriate purifying capability of the catalytic converter, such as athree-way catalyst or the like, disposed in the exhaust gas passage ofthe internal combustion engine (e.g., see Japanese laid-open patentpublication No. 11-324767 or U.S. Pat. No. 6,188,953, and Japaneselaid-open patent publication No. 2000-179385 or U.S. Pat. No.6,230,486).

According to these techniques, an exhaust system ranging from a positionupstream of the catalytic converter to the O₂ sensor disposed downstreamof the catalytic converter is an object to be controlled which has aninput quantity represented by the air-fuel ratio of the exhaust gas thatenters the catalytic converter and an output quantity represented by theoutput of the O₂ sensor. A manipulated variable which determines theinput quantity of the exhaust system, e.g., a target value for the inputquantity of the exhaust system, is sequentially generated by a feedbackcontrol process, or specifically an adaptive sliding mode controlprocess, for converging the output of the O₂ sensor to the target value,and the air-fuel ratio of the air-fuel mixture to be combusted by theinternal combustion engine is controlled depending on the manipulatedvariable.

Generally, the exhaust system including the catalytic converter has arelatively long dead time. The dead time of a system for generating anactual input quantity of the exhaust system from the manipulatedvariable, i.e., an air-fuel ratio manipulating system comprising theinternal combustion engine, etc., is generally smaller than the deadtime of the exhaust system, but may become relatively long depending onthe operating state of the internal combustion engine. These dead timestend to present an obstacle to efforts to smoothly controlling theoutput of the O₂ sensor at the target value.

According to the above techniques, data representing an estimated valueof the output of the O₂ sensor after the dead time of the exhaustsystem, or after a total dead time which is the sum of the dead time ofthe exhaust system and the dead time of the air-fuel ratio manipulatingsystem, is sequentially calculated according to a predeterminedestimating algorithm that is constructed based on a predetermined modelof the exhaust system, etc. In the feedback control process forgenerating the manipulated variable, the estimated value of the outputof the O₂ sensor is used to generate the manipulated variable. That is,the manipulated variable is generated to converge the estimated value tothe target value.

According to the above techniques, the value of a predeterminedparameter of the model of the exhaust system which serves as a basis forthe estimating algorithm is sequentially identified using sampled dataof the output of an air-fuel ratio sensor disposed upstream of thecatalytic converter and the output of the O₂ sensor. The estimatingalgorithm uses the identified value of the parameter of the model of theexhaust system to estimate the output of the O₂ sensor.

By performing the above control procedure, the above techniques cancompensate for the effect of the dead times of the exhaust system andthe air-fuel ratio manipulating system and also for the effect ofbehavioral changes of the exhaust system, and stably and smoothlyperform the control process for converging the output of the O₂ sensorto the target value, or stated otherwise, the control process forachieving an appropriate purifying capability of the catalyticconverter.

According to the above techniques, basically, the dead times of theexhaust system and the dead time of the air-fuel ratio manipulatingsystem are regarded as of constant values, and preset fixed dead timesare used as the values of those dead times. In the process of estimatingthe output of the O₂ sensor, the output of the O₂ sensor after thepreset dead time of the exhaust system, and the output of the O₂ sensorafter the total preset fixed dead time which is the sum of the presetdead time of the exhaust system and the preset dead time of the air-fuelratio manipulating system are sequentially estimated.

The inventors of the present application have found that the actual deadtimes of the exhaust system and the air-fuel ratio manipulating systemvary depending on the state, such as the rotational speed, of theinternal combustion engine. Particularly, the range in which the deadtime of the exhaust system is variable may become relatively largedepending on the operating state of the internal combustion engine.Consequently, depending on the operating state of the internalcombustion engine, the estimated value of the output of the O₂ sensormay have a large error with respect to the output of the O₂ sensor afterthe actual dead time. In the process of identifying the parameter of themodel of the exhaust system, the identified value of the parameter mayvary largely due to an error between the preset dead time of the modeland the actual dead time thereof, i.e., a modeling error relative to adead time element of the exhaust system, possibly resulting in anincreased error of the estimated value of the output of the O₂ sensorwhich is determined using the identified value.

According to the above techniques, since a highly stable control processsuch as an adaptive sliding mode control process is used as the feedbackcontrol process for generating the manipulated variable, it basically ispossible to avoid a situation where the stability of the control processfor converging the output of the O₂ sensor to the target value wouldsignificantly be impaired.

In circumstances where the error of the estimated value of the output ofthe O₂ sensor is relatively large, however, when the manipulatedvariable is generated using the estimated value and the air-fuel ratioof the air-fuel mixture is manipulated depending on the manipulatedvariable, the output of the O₂ sensor tends to vary with respect to thetarget value, and the quick response of the control process convergingthe output of the O₂ sensor to the target value is liable to be lowered.

The applicant of the present application has proposed a technique forvariably setting the preset dead time of the air-fuel manipulatingsystem depending on the rotational speed, etc. of the internalcombustion engine in view of the fact that the actual dead time of theair-fuel manipulating system changes depending on the rotational speed,etc. of the internal combustion engine, as disclosed in Japaneselaid-open patent publication No. 11-324767 or U.S. Pat. No. 6,188,953.The proposed technique, however, does not take into account the factthat the dead time of the exhaust system, which affects the control ofthe output of the O₂ sensor more largely than the dead time of theair-fuel ratio manipulating system, changes depending on the operatingstate of the internal combustion engine, and has a predetermined fixedvalue as the preset dead time of the exhaust system. Therefore, thetechnique disclosed in the above publication also causes the abovedrawbacks.

The present invention has been made in view of the above background. Itis an object of the present invention to provide an apparatus for and amethod of controlling the air-fuel ratio of an internal combustionengine to compensate for the effect of the dead times of an exhaustsystem including a catalytic converter and an air-fuel ratiomanipulating system including the internal combustion engine and henceto increase the purifying capability of the catalytic converter in asystem for manipulating the air-fuel ratio to converge the output of anexhaust gas sensor such as an O₂ sensor or the like disposed downstreamof the catalytic converter to a predetermined target value to achieve anappropriate purifying capability of the catalytic converter. It is alsoan object of the present invention to provide a recording medium storinga program for controlling an air-fuel ratio appropriately with acomputer.

DISCLOSURE OF THE INVENTION

According to the findings of the inventors of the present application,the actual dead time of an exhaust system including a catalyticconverter is closely related particularly to the flow rate of an exhaustgas supplied to the catalytic converter such that the actual dead timeof the exhaust system is longer as the flow rate of the exhaust gas issmaller (see the solid-line curve c in FIG. 4). Furthermore, the deadtime of an air-fuel ratio manipulating system including an internalcombustion engine is also highly correlated to the flow rate of theexhaust gas such that the dead time of the air-fuel ratio manipulatingsystem is longer as the flow rate of the exhaust gas is smaller (see thesolid-line curve d in FIG. 4).

The present invention is based on the above phenomenon, and has a firstaspect capable of compensating for the effect of the dead time of anexhaust system including a catalytic converter and a second aspectcapable of compensating for the effect of the dead time of an air-fuelratio manipulating system including an internal combustion engine, inaddition to the dead time of the exhaust system.

According to the first aspect of the present invention, there isprovided an apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, estimating means for sequentially generating datarepresentative of an estimated value of an output of said exhaust gassensor after a set dead time which is set as a dead time of an exhaustsystem ranging from a position upstream of said catalytic converter tosaid exhaust gas sensor and including said catalytic converter,manipulated variable generating means for generating a manipulatedvariable to determine an air-fuel ratio of the exhaust gas which enterssaid catalytic converter to converge the output of said exhaust gassensor to a predetermined target value, using the data generated by saidestimating means, and air-fuel ratio manipulating means for manipulatingthe air-fuel ratio of an air-fuel mixture to be combusted by theinternal combustion engine depending on the manipulated variable, saidapparatus comprising flow rate data generating means for sequentiallygenerating data representative of a flow rate of the exhaust gassupplied to the catalytic converter, and dead time setting means forvariably setting a value of said set dead time depending on the value ofthe data generated by said flow rate data generating means, whereinthere is established a predetermined model of said exhaust system forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of said exhaust gas sensor via a deadtime element and a response delay element of said set dead time from theair-fuel ratio of the exhaust gas which enters said catalytic converter,further comprising identifying means for sequentially identifying thevalue of a predetermined parameter of said model using the value of theset dead time set by said dead time setting means, wherein saidestimating means generates the data representative of the estimatedvalue of the output of said exhaust gas sensor using the identifiedvalue of said parameter determined by said identifying means, accordingto a predetermined estimating algorithm which is constructed based onthe model of said exhaust system, and wherein said identifying meansdetermines the identified value of the parameter of the model of saidexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data generated by saidflow rate data generating means.

According to the first aspect of the present invention, there isprovided a method of controlling the air-fuel ratio of an internalcombustion engine, comprising the steps of sequentially generating datarepresentative of an estimated value of an output of an exhaust gassensor disposed downstream of a catalytic converter disposed in anexhaust passage of the internal combustion engine, for detecting theconcentration of a particular component in an exhaust gas which haspassed through the catalytic converter, after a set dead time which isset as a dead time of an exhaust system ranging from a position upstreamof said catalytic converter to said exhaust gas sensor and includingsaid catalytic converter, and generating a manipulated variable todetermine an air-fuel ratio of the exhaust gas which enters saidcatalytic converter to converge the output of said exhaust gas sensor toa predetermined target value, using the data representative of theestimated value, wherein the air-fuel ratio of an air-fuel mixture to becombusted by the internal combustion engine is manipulated depending onthe manipulated variable said method comprising the steps ofsequentially generating data representative of a flow rate of theexhaust gas supplied to the catalytic converter, and variably setting avalue of said set dead time depending on the value of the datarepresentative of the flow rate of the exhaust gas, wherein there isestablished a predetermined model of said exhaust system for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of said exhaust gas sensor via a dead time elementand a response delay element of said set dead time from the air-fuelratio of the exhaust gas which enters said catalytic converter, furthercomprising the step of sequentially identifying the value of apredetermined parameter of said model using the value of said set deadtime, wherein said step of generating data representative of theestimated value of the output of the exhaust gas sensor generates thedata representative of the estimated value of the output of said exhaustgas sensor using the identified value of said parameter, according to apredetermined estimating algorithm which is constructed based on themodel of said exhaust system, and wherein said step of identifying theparameter of the model of said exhaust system determines the identifiedvalue of the parameter of the model of said exhaust system by limitingthe identified value to a value within a predetermined range dependingon the value of the data representative of the flow rate of the exhaustgas supplied to said catalytic converter.

According to the first aspect of the present invention, there isprovided a recording medium readable by a computer and storing anair-fuel ratio control program for enabling said computer to perform aprocess of sequentially generating data representative of an estimatedvalue of an output of an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, after a set dead time which is set as a dead time of anexhaust system ranging from a position upstream of said catalyticconverter to said exhaust gas sensor and including said catalyticconverter, a process of generating a manipulated variable to determinean air-fuel ratio of the exhaust gas which enters said catalyticconverter to converge the output of said exhaust gas sensor to apredetermined target value, using the data representative of theestimated value, and a process of manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable, said air-fuel ratio controlprogram comprising a program of enabling the computer to perform aprocess of sequentially generate data representative of a flow rate ofthe exhaust gas supplied to the catalytic converter, and variablysetting a value of said set dead time depending on the value of the datarepresentative of the flow rate of the exhaust gas, wherein there isestablished a predetermined model of said exhaust system for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of said exhaust gas sensor via a dead time elementand a response delay element of said set dead time from the air-fuelratio of the exhaust gas which enters said catalytic converter, saidair-fuel ratio control program includes a program for enabling thecomputer to perform a process of sequentially identifying the value of apredetermined parameter of said model using the value of the set deadtime set by said dead time setting means, wherein the program of saidair-fuel ratio control program for generating the data representative ofthe estimated value of the output of the exhaust gas sensor enables thecomputer to generate the data representative of the estimated value ofthe output of said exhaust gas sensor using the identified value of saidparameter, according to a predetermined estimating algorithm which isconstructed based on the model of said exhaust system, and wherein theprogram of said air-fuel ratio control program for identifying theparameter of the model of said exhaust system determines the identifiedvalue of the parameter of the model of said exhaust system by limitingthe identified value to a value within a predetermined range dependingon the value of the data representative of the flow rate of the exhaustgas supplied to said catalytic converter.

According to the first aspect of the present invention, the value of theset dead time of the exhaust system established depending on the valueof the data representiative of the flow rate of the exhaust gas suppliedto the catalytic converter. Therefore, the set dead time can be broughtinto conformity with the actual dead time of the exhaust system withaccuracy. Basically, the set dead time is established such that it isgreater as the flow rate of the exhaust gas supplied to the catalyticconverter is smaller.

According to the first aspect of the present invention, the datarepresentative of the estimated value of the output of the exhaust gassensor after the set dead time is sequentially generated using the setdead time thus established. Therefore, the estimated value of the outputof the exhaust gas sensor which is represented by the data is highlyreliable as the estimated value of the output of the exhaust gas sensorafter the actual dead time of the exhaust system, and the accuracy ofthe estimated value is increased. Using the highly reliable data as thedata representative of the estimated value of the output of the exhaustgas sensor after the actual dead time of the exhaust system, themanipulated variable is generated, and the air-fuel ratio of theair-fuel mixture to be combusted by the internal combustion engine ismanipulated depending on the manipulated variable. Therefore, the effectof the dead time can appropriately be compensated for depending on thelength of the actual dead time of the exhaust system, and hence theaccuracy and quick response of the control process for converging theoutput of the exhaust gas sensor to the target value is increased. As aresult, the purifying capability of the catalytic converter isincreased.

According to the second aspect of the present invention, there isprovided an apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, manipulated variable generating means for sequentiallygenerating a manipulated variable to determine an air-fuel ratio of theexhaust gas which enters the catalytic converter to converge an outputof the exhaust gas sensor to a predetermined target value, air-fuelratio manipulating means for manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable, and estimating means forsequentially generating data representative of an estimated value of theoutput of the exhaust gas sensor after a total set dead time which isthe sum of a first set dead time and a second set dead time, the firstset dead time being set as a dead time of an exhaust system ranging froma position upstream of the catalytic converter to the exhaust gas sensorand including the catalytic converter, said second set dead time beingset as a dead time of an air-fuel ratio manipulating system comprisingsaid air-fuel ratio manipulating means and said internal combustionengine, wherein said manipulated variable generating means generatessaid manipulated variable using the data generated by said estimatingmeans, said apparatus comprising flow rate data generating means forsequentially generating data representative of a flow rate of theexhaust gas supplied to the catalytic converter, and dead time settingmeans for variably setting values of said first set dead time and saidsecond set dead time depending on the value of the data generated bysaid flow rate data generating means, wherein there is established apredetermined model of said exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of said exhaust gas sensor via a dead time element and a responsedelay element of said first set dead time from the air-fuel ratio of theexhaust gas which enters said catalytic converter, further comprisingidentifying means for sequentially identifying the value of apredetermined parameter of said model using the value of the first setdead time set by said dead time setting means, wherein said estimatingmeans generates the estimated value of the output of said exhaust gassensor using the identified value of said parameter determined by saididentifying means, according to a predetermined estimating algorithmwhich is constructed based on the model of said exhaust system and apredetermined model of said air-fuel ratio manipulating means forexpressing a behavior of the air-fuel ratio manipulating means which isregarded as a system for generating the air-fuel ratio detected by saidair-fuel ratio sensor from said manipulated variable via a dead timeelement of said second set dead time, and wherein said identifying meansdetermines the identified value of the parameter of the model of saidexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data generated by saidflow rate data generating means.

According to the second aspect of the present invention, there isprovided a method of controlling the air-fuel ratio of an internalcombustion engine, comprising the steps of sequentially generating amanipulated variable to determine an air-fuel ratio of the exhaust gaswhich enters said catalytic converter to converge an output of anexhaust gas sensor, which is disposed downstream of a catalyticconverter disposed in an exhaust passage of the internal combustionengine, for detecting the concentration of a particular component in anexhaust gas which has passed through the catalytic converter, to apredetermined target value, manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable, and sequentially generating datarepresentative of an estimated value of the output of said exhaust gassensor after a total set dead time which is the sum of a first set deadtime and a second set dead time, said first set dead time being set as adead time of an exhaust system ranging from a position upstream of saidcatalytic converter to said exhaust gas sensor and including saidcatalytic converter, said second set dead time being set as a dead timeof an air-fuel ratio manipulating system comprising said air-fuel ratiomanipulating means and said internal combustion engine, wherein saidstep of generating said manipulated variable uses the datarepresentative of the estimated value of the output of the exhaust gassensor in order to generate said manipulated variable, said methodcomprising the steps of by sequentially generating data representativeof a flow rate of the exhaust gas supplied to the catalytic converter,and variably setting values of said first set dead time and said secondset dead time depending on the value of the data representative of theflow rate of the exhaust gas, wherein there is established apredetermined model of said exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of said exhaust gas sensor via a dead time element and a responsedelay element of said first set dead time from the air-fuel ratio of theexhaust gas which enters said catalytic converter, further comprisingthe step of sequentially identifying the value of a predeterminedparameter of said model using the value of said first set dead time,wherein said step of generating the data representative of the estimatedvalue of the output of said exhaust gas sensor generates the estimatedvalue of the output of said exhaust gas sensor using the identifiedvalue of said parameter of the model of said exhaust system, accordingto a predetermined estimating algorithm which is constructed based onthe model of said exhaust system and a predetermined model of saidair-fuel ratio manipulating means for expressing a behavior of theair-fuel ratio manipulating means which is regarded as a system forgenerating the air-fuel ratio detected by said air-fuel ratio sensorfrom said manipulated variable via a dead time element of said secondset dead time, and said step of identifying the parameter of the modelof said exhaust system determines the identified value of the parameterof the model of said exhaust system by limiting the identified value toa value within a predetermined range depending on the value of the datarepresentative of the flow rate of the exhaust gas supplied to saidcatalytic converter.

According to the second aspect of the present invention, there isprovided a recording medium readable by a computer and storing anair-fuel ratio control program for enabling said computer to perform aprocess of sequentially generating a manipulated variable to determinean air-fuel ratio of the exhaust gas which enters the catalyticconverter to converge an output of an exhaust gas sensor, which isdisposed downstream of a catalytic converter disposed in an exhaustpassage of the internal combustion engine, for detecting theconcentration of a particular component in an exhaust gas which haspassed through the catalytic converter, to a predetermined target value,a process of manipulating the air-fuel ratio of an air-fuel mixture tobe combusted by the internal combustion engine depending on themanipulated variable, and a process of sequentially generating datarepresentative of an estimated value of the output of the exhaust gassensor after a total set dead time which is the sum of a first set deadtime and a second set dead time, the first set dead time being set as adead time of an exhaust system ranging from a position upstream of thecatalytic converter to the exhaust gas sensor and including thecatalytic converter, the second set dead time being set as a dead timeof an air-fuel ratio manipulating system comprising the air-fuel ratiomanipulating means and the internal combustion engine, wherein theprogram of the air-fuel ratio control program for generating themanipulated variable is constructed of an algorithm for generating themanipulated variable using the data representative of the estimatedvalue of the output of the exhaust gas sensor, the air-fuel ratiocontrol program comprising a program for enabling the computer toperform a process of sequentially generating data representative of aflow rate of the exhaust gas supplied to the catalytic converter, andvariably setting values of said first set dead time and said second setdead time depending on the value of the data representative of the flowrate of the exhaust gas, wherein there is established a predeterminedmodel of said exhaust system for expressing a behavior of the exhaustsystem which is regarded as a system for generating the output-of saidexhaust gas sensor via a dead time element and a response delay elementof said first set dead time from the air-fuel ratio of the exhaust gaswhich enters said catalytic converter, said air-fuel ratio controlprogram includes a program for enabling the computer to perform aprocess of sequentially identifying the value of a predeterminedparameter of said model using the value of said first set dead time,wherein the program of said air-fuel ratio control program forgenerating the data representative of the estimated value of the outputof said exhaust gas sensor enables the computer to generate theestimated value of the output of said exhaust gas sensor using theidentified value of said parameter of the model of said exhaust system,according to a predetermined estimating algorithm which is constructedbased on the model of said exhaust system and a predetermined model ofsaid air-fuel ratio manipulating means for expressing a behavior of theair-fuel ratio manipulating means which is regarded as a system forgenerating the air-fuel ratio detected by said air-fuel ratio sensorfrom said manipulated variable via a dead time element of said secondset dead time, and wherein the program of said air-fuel ratio controlprogram for identifying the parameter of the model of said exhaustsystem determines the identified value of the parameter of the model ofsaid exhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data representative ofthe flow rate of the exhaust gas supplied to said catalytic converter.

According to the second aspect of the present invention, the first setdead time as the dead time of the exhaust system and the second set deadtime as the dead time of the air-fuel ratio manipulating system areestablished depending on the value of the data representative of theflow rate of the exhaust gas supplied to the catalytic converter.Therefore, the first and second set dead times can be brought intoconformity with the actual dead time of the exhaust system and theactual dead time of the air-fuel ratio manipulating system withaccuracy. Basically, the first set dead time and the second set deadtime are established such that they are greater as the flow rate of theexhaust gas supplied to the catalytic converter is smaller.

According to the second aspect of the present invention, the datarepresentative of the estimated value of the output of the exhaust gassensor after a total set dead time which is the sum of the first setdead time and the sound set dead time is generated using the first andsecond set dead times thus established. Therefore, the estimated valueof the output of the exhaust gas sensor which is represented by the datais highly reliable as the estimated value of the output of the exhaustgas sensor after the actual total dead time which is the sum of theactual dead time of the exhaust system and the actual dead time of theair-fuel ratio manipulating system, and the accuracy of the estimatedvalue is increased. According to the second aspect of the presentinvention, using the highly reliable data as the data representative ofthe estimated value of the output of the exhaust gas sensor after theactual total dead time, the manipulated variable is generated, and theair-fuel ratio of the air-fuel mixture to be combusted by the internalcombustion engine is manipulated depending on the manipulated variable.Therefore, the effect of the dead times can appropriately be compensatedfor depending on the length of the actual dead times of the exhaustsystem and the air-fuel ratio manipulating system, and hence theaccuracy and quick response of the control process for converging theoutput of the exhaust gas sensor to the target value is increased. As aresult, the purifying capability of the catalytic converter isincreased.

The first aspect of the present invention is suitable when the dead timeof the air-fuel ratio manipulating system is sufficiently smaller thanthe dead time of the exhaust system, and the second aspect of thepresent invention is suitable when the dead time of the air-fuel ratiomanipulating system is relatively long. According to the first andsecond aspects of the present invention, in order to generate thesuitable manipulated variable for compensating for the effect of thedead time of the exhaust system or the effect of the total dead timewhich is the sum of the dead time of the exhaust system and the deadtime of the air-fuel ratio manipulating system, it is preferable togenerate the manipulated variable according to a feedback controlprocess for converging the estimated value of the output of the exhaustgas sensor to the target value. The manipulated variable may be, forexample, a target value for the air-fuel ratio (target air-fuel ratio)of the exhaust gas that enters the catalytic converter or a correctivequantity for the amount of fuel supplied to the internal combustionengine. If the manipulated variable is the target air-fuel ratio, thenan air-fuel sensor for detecting the air-fuel ratio of the exhaust gasthat enters the catalytic converter is preferably disposed upstream ofthe catalytic converter, and the air-fuel ratio of an air-fuel mixtureto be combusted by the internal combustion engine is preferablymanipulated according to a feedback control process for converting theoutput of the air-fuel ratio sensor (detected value of the air-fuelratio) to the target air-fuel ratio.

In the apparatus for controlling the air-fuel ratio according to thefirst aspect of the present invention, preferably, there is establisheda predetermined model of the exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of the exhaust gas sensor via a dead time element and a responsedelay element of the set dead time from the air-fuel ratio of theexhaust gas which enters the catalytic converter, the apparatus furthercomprising identifying means for sequentially identifying the value of apredetermined parameter of the model using the value of the set deadtime set by the dead time setting means, wherein the estimating meansgenerates the data representative of the estimated value of the outputof the exhaust gas sensor using the identified value of the parameterdetermined by the identifying means, according to a predeterminedestimating algorithm which is constructed based on the model of theexhaust system.

In the method of controlling the air-fuel ratio according to the firstaspect of the present invention, similarly, there is established apredetermined model of the exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of the exhaust gas sensor via a dead time element and a responsedelay element of the set dead time from the air-fuel ratio of theexhaust gas which enters the catalytic converter, the method furthercomprising the step of sequentially identifying the value of apredetermined parameter of the model using the value of the set deadtime, wherein the step of generating data representative of theestimated value of the output of the exhaust gas sensor generates thedata representative of the estimated value of the output of the exhaustgas sensor using the identified value of the parameter, according to apredetermined estimating algorithm which is constructed based on themodel of the exhaust system.

In the recording medium storing the air-fuel ratio control programaccording to the first aspect of the present invention, preferably,there is established a predetermined model of the exhaust system forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of the exhaust gas sensor via a deadtime element and a response delay element of the set dead time from theair-fuel ratio of the exhaust gas which enters the catalytic converter,wherein the air-fuel ratio control program includes a program forenabling the computer to perform a process of sequentially identifyingthe value of a predetermined parameter of the model using the value ofthe set dead time, and the program of the air-fuel ratio control programfor generating the data representative of the estimated value of theoutput of the exhaust gas sensor enables the computer to generate thedata representative of the estimated value of the output of the exhaustgas sensor using the identified value of the parameter, according to analgorithm which is constructed based on the model of the exhaust system.

Specifically, an estimating algorithm capable of appropriatelygenerating the data representative of the estimated value of the outputof the exhaust gas sensor after the set dead time of the exhaust systemcan be constructed based on the model of the exhaust system establishedas described above. The estimating algorithm requires not only the valueof the set dead time, but also the value of a predetermined parameter ofthe model of the exhaust system (a parameter to be set to a certainvalue for determining the behavior of the model, e.g., a coefficientparameter relative to the dead time element and the response delayelement of the model of the exhaust system). The value of the parameterof the model that matches the actual behavior of the exhaust systemgenerally changes depending on changes in the behavior andcharacteristics of the exhaust system. According to the presentinvention, therefore, the value of the parameter of the model issequentially identified in order to compensate for the affect of changesin the behavior and characteristics of the exhaust system. In theidentifying process, the set dead time established by the dead timesetting means, i.e., the value of the set dead time that well matchesthe actual dead time of the exhaust system with accuracy, is used as thedead time of the dead time element of the model of the exhaust system,so that matching between the behavior of the model of the exhaust modeland the behavior of the actual exhaust system is increased, thusincreasing the reliability of the identified value of the parameter ofthe model. When the estimating means generates the data representativeof the output of the exhaust gas sensor according to the estimatingalgorithm using the identified value of the parameter according to theestimating algorithm, it is possible to generate highly reliable data asthe data representative of the estimated value of the output of theexhaust gas sensor after the actual dead time of the exhaust system,irrespective of changes in the behavior of the exhaust system. As aconsequence, the accuracy and quick response of the control process forconverging the output of the exhaust gas sensor to the target value isincreased, and hence the purifying capability of the catalytic converteris increased.

In the apparatus for controlling the air-fuel ratio according to thesecond aspect of the present invention, preferably, there is establisheda predetermined model of the exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of the exhaust gas sensor via a dead time element and a responsedelay element of the first set dead time from the air-fuel ratio of theexhaust gas which enters the catalytic converter, the apparatus furthercomprising identifying means for sequentially identifying the value of apredetermined parameter of the model using the value of the first setdead time set by the dead time setting means, wherein the estimatingmeans generates the estimated value of the output of the exhaust gassensor using the identified value of the parameter determined by theidentifying means, according to a predetermined estimating algorithmwhich is constructed based on the model of the exhaust system and apredetermined model of the air-fuel ratio manipulating means forexpressing a behavior of the air-fuel ratio manipulating means which isregarded as a system for generating the air-fuel ratio detected by theair-fuel ratio sensor from the manipulated variable via a dead timeelement of the second set dead time.

In the method of controlling the air-fuel ratio according to the secondaspect of the present invention, similarly, there is established apredetermined model of the exhaust system for expressing a behavior ofthe exhaust system which is regarded as a system for generating theoutput of the exhaust gas sensor via a dead time element and a responsedelay element of the first set dead time from the air-fuel ratio of theexhaust gas which enters the catalytic converter, the method furthercomprising the step of sequentially identifying the value of apredetermined parameter of the model using the value of the first setdead time set by the dead time setting means, wherein the step ofgenerating the data representative of the estimated value of the outputof the exhaust gas sensor generates the estimated value of the output ofthe exhaust gas sensor using the identified value of the parameter ofthe model of the exhaust system, according to a predetermined estimatingalgorithm which is constructed based on the model of the exhaust systemand a predetermined model of the air-fuel ratio manipulating means forexpressing a behavior of the air-fuel ratio manipulating means which isregarded as a system for generating the air-fuel ratio detected by theair-fuel ratio sensor from the manipulated variable via a dead timeelement of the second set dead time.

In the recording medium storing the air-fuel ratio control programaccording to the second aspect of the present invention, preferably,there is established a predetermined model of the exhaust system forexpressing a behavior of the exhaust system which is regarded as asystem for generating the output of the exhaust gas sensor via a deadtime element and a response delay element of the first set dead timefrom the air-fuel ratio of the exhaust gas which enters the catalyticconverter, wherein the air-fuel ratio control program includes a programfor enabling the computer to perform a process of sequentiallyidentifying the value of a predetermined parameter of the model usingthe value of the first set dead time set by the dead time setting means,wherein the program of the air-fuel ratio control program for generatingthe data representative of the estimated value of the output of theexhaust gas sensor enables the computer to generate the estimated valueof the output of the exhaust gas sensor using the identified value ofthe parameter of the model of the exhaust system, according to analgorithm which is constructed based on the model of the exhaust systemand a predetermined model of the air-fuel ratio manipulating means forexpressing a behavior of the air-fuel ratio manipulating means which isregarded as a system for generating the air-fuel ratio detected by theair-fuel ratio sensor from the manipulated variable via a dead timeelement of the second set dead time.

According to the second aspect of the present invention, the estimatingalgorithm capable of appropriately generating the data representative ofthe estimated value of the output of the exhaust gas sensor after thetotal set dead time can be constructed based on the model of the exhaustsystem and the model of the air-fuel ratio manipulating means. As withthe first aspect, the estimating algorithm requires not only the valuesof the first and second set dead times, but also the value of apredetermined parameter of the model of the exhaust system. According tothe present invention, as with the first aspect, the value of theparameter is sequentially identified in order to compensate for theaffect of changes in the behavior and characteristics of the exhaustsystem. In the identifying process, the first set dead time establishedby the dead time setting means (that well matches the actual dead timeof the exhaust system with accuracy) is used as the dead time of thedead time element of the model of the exhaust system in order todetermine the identified value of the model of the exhaust system.Therefore, matching between the behavior of the model of the exhaustmodel and the behavior of the actual exhaust system is increased, thusincreasing the reliability of the identified value of the parameter ofthe model of the exhaust system. When the estimating means generates thedata representative of the output of the exhaust gas sensor according tothe estimating algorithm using the identified value of the parameteraccording to the estimating algorithm, it is possible to generate highlyreliable data as the data representative of the estimated value of theoutput of the exhaust gas sensor after the actual total dead time, whichis the sum of the actual dead time of the exhaust system and the actualdead time of the air-fuel ratio manipulating system, irrespective ofchanges in the behavior of the exhaust system. As a consequence, theaccuracy and quick response of the control process for converging theoutput of the exhaust gas sensor to the target value is increased, andhence the purifying capability of the catalytic converter is increased.

In the estimating algorithm according to the first aspect of the presentinvention, more specifically, it is possible to appropriately generatethe data representative of the estimated value of the output of theexhaust gas sensor, using the data (time-series data) of the air-fuelratio of the exhaust gas that enters the catalytic converter (which mayhereinafter be referred to as “upstream-of-catalyst air-fuel ratio”),the data of the output of the exhaust gas sensor, the value of the setdead time of the exhaust system, and the identified value of theparameter of the model of the exhaust system, for example. Likewise, inthe estimating algorithm according to the second aspect of the presentinvention, it is possible to appropriately generate the datarepresentative of the estimated value of the output of the exhaust gassensor, using the data (time-series data) representative of theupstream-of-catalyst air-fuel ratio, the data of the output of theexhaust gas sensor, the values of the first and second set dead times,and the identified value of the parameter of the model of the exhaustsystem, for example. In either of the aspects, since theupstream-of-catalyst air-fuel ratio is determined by the manipulatedvariable, the data of the manipulated variable can be used as the datarepresentative of the upstream-of-catalyst air-fuel ratio. For makingthe data representative of the upstream-of-catalyst air-fuel ratio moreadequate, it is preferable to provide an air-fuel ratio sensor fordetecting the upstream-of-catalyst air-fuel ratio upstream of thecatalytic converter and use the data of an output of the air-fuel ratiosensor, or to use both the data of the output of the air-fuel ratiosensor and the data of the manipulated variable.

According to the first and second aspects of the present invention, morespecifically, it is possible to identify the parameter of the model ofthe exhaust system according the algorithm of a sequential method ofweighted least squares, using the data representative of theupstream-of-catalyst air-fuel ratio and the value of the set dead timeof the model of the exhaust system (the first set dead time in thesecond aspect). As with the estimating algorithm, the data of themanipulated variable can be used as the data representative of theupstream-of-catalyst air-fuel ratio, but it is preferable to provide anair-fuel ratio sensor for detecting the upstream-of-catalyst air-fuelratio upstream of the catalytic converter and use the data of an outputof the air-fuel ratio sensor

The model of the exhaust system according to the first and secondaspects of the present invention should preferably be a model whichexpresses the data of the output of the exhaust gas sensor in each givencontrol cycle with the data of the output of the exhaust gas sensor in apast control cycle prior to the control cycle and the datarepresentative of the upstream-of-catalyst air-fuel ratio (the data ofthe output of the air-fuel ratio sensor, the data of the manipulatedvariable, or the like) in a control cycle prior to the set dead time ofthe exhaust system, for example. Stated otherwise, the model shouldpreferably be an autoregressive model where the upstream-of-catalystair-fuel ratio as an input quantity to the exhaust system has a deadtime (the set dead time of the exhaust system), for example. Theparameter of the model comprises a coefficient relative to the data(autoregressive term relative to an output quantity of the exhaustsystem) of the output of the exhaust gas sensor in the past controlcycle, or a coefficient relative to the data (input quantity to theexhaust system) representative of the upstream-of-catalyst air-fuelratio. The model of the air-fuel ratio manipulating system in the secondaspect is expressed as a system where the upstream-of-catalyst air-fuelratio in each given control cycle coincides with the air-fuel ratio thatis determined by the manipulated variable prior to the second set deadtime.

In the apparatus for controlling the air-fuel ratio according to thefirst and second aspects of the present invention to identify theparameter of the model of the exhaust model, the identifying meansdetermines the identified value of the parameter of the model of theexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data generated by theflow rate data generating means.

In the method of controlling the air-fuel ratio according to the firstand second aspects of the present invention, the step of identifying theparameter of the model of the exhaust system determines the identifiedvalue of the parameter of the model of the exhaust system by limitingthe identified value to a value within a predetermined range dependingon the value of the data generated by the flow rate data generatingmeans.

In the recording medium storing the air-fuel ratio control programaccording to the first and second aspects of the present invention,preferably, the program of the air-fuel ratio control program foridentifying the parameter of the model of the exhaust system determinesthe identified value of the parameter of the model of the exhaust systemby limiting the identified value to a value within a predetermined rangedepending on the value of the data generated by the flow rate datagenerating means.

Specifically, the identified value of the parameter which is suitablefor generating the manipulated variable capable of converging the outputof the exhaust gas sensor smoothly to the target value in order toincrease the accuracy of the data representative of the estimated valueof the output of the exhaust gas sensor is generally affected by theactual dead time of the exhaust system. According to the first andsecond aspects of the present invention, the identified value is limitedto a value within a predetermined range depending on the value of thedata representative of the flow rate of the exhaust gas supplied to thecatalytic converter. It is thus possible to determine the identifiedvalue suitable for generating the manipulated variable capable ofconverging the output of the exhaust gas sensor smoothly to the targetvalue.

If there are a plurality of parameters to be identified of the model ofthe exhaust system, then the predetermined range within which to limitthe identified values of those parameters may be a range for each of theidentified values of those parameters or a range for a combination ofthe identified values of those parameters. For example, if the model ofthe exhaust system is an autoregressive model and its autoregressiveterms include primary and secondary autoregressive terms (whichcorrespond to the response delay element of the exhaust system), then itis preferable to limit a combination of the identified values of twoparameters relative to the respective autoregressive terms within apredetermined range (specifically, a predetermined area on a coordinateplane having the values of the two parameters as representing twocoordinate axes). The identified value of a parameter relative to theupstream-of-catalyst air-fuel ratio of the autoregressive model shouldpreferably be limited to a value within a predetermined range (a rangehaving upper and lower limit values).

In the apparatus for controlling the air-fuel ratio according to thefirst and second aspects of the present invention to identify theparameter of the model of the exhaust model, preferably, the identifyingmeans comprises means for identifying the value of the parameteraccording to an algorithm for minimizing an error between the output ofthe exhaust gas sensor in the model of the exhaust system and an actualoutput of the exhaust gas sensor, e.g., the algorithm of a method ofweighted least squares, and the apparatus further comprises means forvariably setting the value of a weighted parameter of the algorithmdepending on the value of the data generated by the flow rate datagenerating means.

In the method of controlling the air-fuel ratio according to the firstand second aspects of the present invention, similarly, the step ofidentifying the parameter of the model of the exhaust system identifiesthe value of the parameter according to an algorithm for minimizing anerror between the output of the exhaust gas sensor in the model of theexhaust system and an actual output of the exhaust gas sensor, andvariably sets the value of a weighted parameter of the algorithmdepending on the value of the data representative of the flow rate ofthe exhaust gas.

In the recording medium storing the air-fuel ratio control programaccording to the first and second aspects of the present invention,preferably, the program of the air-fuel ratio control program foridentifying the parameter of the model of the exhaust system identifiesthe value of the parameter according to an algorithm for minimizing anerror between the output of the exhaust gas sensor in the model of theexhaust system and an actual output of the exhaust gas sensor, andvariably sets the value of a weighted parameter of the algorithmdepending on the value of the data representing the flow rate of theexhaust gas.

Specifically, the algorithm for identifying the parameter of the modelof the exhaust system may be any one of various specific algorithmsincluding a method of least squares, a method of weighted least squares,a fixed gain method, a degressive gain method, etc. According to thefindings of the inventors of the present application, as the actual deadtime of the exhaust system is longer, the identified value of theparameter of the model of the exhaust system is more liable to vary.According to the findings of the inventors of the present application,furthermore, as the flow rate of the exhaust gas is lower, the actualresponse delay element of the exhaust system is longer, and theidentified value of the parameter of the model of the exhaust system ismore liable to suffer an error. The data representative of the estimatedvalue of the output of the exhaust gas sensor is also more liable tosuffer an error, tending to impair the quick response of the process forcontrolling the output of the exhaust gas sensor at the target value.According to the algorithm of the method of weighted least squares, itis possible to suppress variations of the identified value of theparameter of the model of the exhaust system and also to suppress anerror of the identified value by adjusting the value of the weightedparameter.

According to the present invention, an algorithm such as a method ofweighted least squares (an algorithm for minimizing an error between theoutput of the exhaust sensor in the model of the exhaust system and theactual output of the exhaust sensor) is used to identify the value ofthe parameter of the model of the exhaust system. The value of theweighted parameter is variably set depending on the value of the datarepresentative of the flow rate of the exhaust gas supplied to thecatalytic converter. It is thus possible to adjust the value of theweighted parameter in accordance with the actual dead time and responsedelay characteristics of the exhaust system. As a result, it is possibleto suppress variations and errors of the identified value of theparameter of the model of the exhaust system and increase the accuracyof the data representative of the estimated value of the output of theexhaust gas sensor. Hence, the quick response and accuracy of thecontrol process for converging the output of the exhaust gas sensor tothe target value can be increased.

In the apparatus for controlling the air-fuel ratio according to thefirst and second aspects of the present invention to identify theparameter of the model of the exhaust model, preferably, the manipulatedvariable generating means generates the manipulated variable using theidentified value, determined by the identifying means, of the parameterof the model of the exhaust system.

In the method of controlling the air-fuel ratio according to the firstand second aspects of the present invention, similarly, the step ofgenerating the manipulated variable uses the identified value of theparameter of the model of the exhaust system determined by theidentifying means in order to generate the manipulated variable.

In the recording medium storing the air-fuel ratio control programaccording to the first and second aspects of the present invention,preferably, the program of the air-fuel ratio control program forgenerating the manipulated variable is constructed of an algorithm forusing the identified value of the parameter of the model of the exhaustsystem in order to generate the manipulated variable.

With the above arrangement, the manipulated variable can be generated soas to reflect the actual behavior and characteristics of the model ofthe exhaust system. Consequently, it is possible to generate amanipulated variable that is more suitable for converging the output ofthe exhaust gas sensor to the target value. Thus, the quick response andaccuracy of the control process for converging the output of the exhaustgas sensor to the target value can be increased, increasing thepurifying capability of the catalytic converter. In particular, when theidentified value of the parameter of the model of the exhaust system islimited within a predetermined range depending on the datarepresentative of the flow rate of the exhaust gas supplied to thecatalytic converter, or the value of the weighted parameter of thealgorithm of the identifying process is established depending on thedata representative of the flow rate of the exhaust gas, the adequacy ofthe identified value of the parameter of the model of the exhaust systemis increased. Therefore, the above advantages are enhanced.

The feedback control process for generating the manipulated variableshould preferably be an adaptive control process, or more specifically,a sliding mode control process. The sliding mode control process may bean ordinary sliding mode control process based on a control law relativeto an equivalent control input and a reaching law, but should preferablybe an adaptive sliding mode control process with an adaptive law(adaptive algorithm) added to those control laws.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an overall system arrangement of anapparatus for controlling the air-fuel ratio of an internal combustionengine according to a first embodiment of the present invention;

FIG. 2 is a diagram showing output characteristics of an O₂ sensor usedin the apparatus shown in FIG. 1;

FIG. 3 is a block diagram showing a basic arrangement of a targetair-fuel ratio generation processor of the apparatus shown in FIG. 1;

FIG. 4 is a diagram illustrative of a process performed by a dead timesetting means of the target air-fuel ratio generation processor shown inFIG. 3;

FIG. 5 is a diagram illustrative of a process performed by an identifierof the target air-fuel ratio generation processor shown in FIG. 3;

FIG. 6 is a diagram with respect to a sliding mode controller of thetarget air-fuel ratio generation processor shown in FIG. 3;

FIG. 7 is a block diagram showing a basic arrangement of an adaptivecontroller of the apparatus shown in FIG. 1;

FIG. 8 is a flowchart of a processing sequence of an engine-side controlunit (7 b) of the apparatus shown in FIG. 1;

FIG. 9 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 10 is a flowchart of an overall processing sequence of anexhaust-side control unit (7 a) of the apparatus shown in FIG. 1;

FIGS. 11 and 12 are flowcharts of subroutines of the flowchart shown inFIG. 10;

FIGS. 13 through 15 are diagrams illustrating partial processes of theflowchart shown in FIG. 12;

FIG. 16 is a flowchart of a subroutine of the flowchart shown in FIG.12; and

FIG. 17 is a flowchart of a subroutine of the flowchart shown in FIG.10.

BEST MODE FOR CARRYING OUT THE INVENTION

A first embodiment of the present invention will be described below withreference to FIGS. 1 through 17. The present embodiment is an embodimentrelating to the second aspect of the present invention.

FIG. 1 shows in block form an overall system arrangement of an apparatusfor controlling the air-fuel ratio of an internal combustion engineaccording to the present embodiment. As shown in FIG. 1, an internalcombustion engine 1 such as a four-cylinder internal combustion engineis mounted as a propulsion source on an automobile or a hybrid vehicle,for example. When a mixture of fuel and air is combusted in eachcylinder of the internal combustion engine 1, an exhaust gas isgenerated and emitted from each cylinder into a common discharge pipe 2(exhaust passage) positioned near the internal combustion engine 1, fromwhich the exhaust gas is discharged into the atmosphere. Two three-waycatalytic converters 3, 4, each comprising a three-way catalyst forpurifying the exhaust gas, are mounted in the common exhaust pipe 2 atsuccessively downstream locations thereon. The downstream catalyticconverter 4 may be dispensed with.

The system according to the present embodiment serves to control theair-fuel ratio of the internal combustion engine 1 (or more accurately,the air-fuel ratio of the mixture of fuel and air to be combusted by theinternal combustion engine 1) in order to achieve an optimum purifyingcapability of the catalytic converter 3. In order to perform the abovecontrol process, the system according to the present embodiment has anair-fuel ratio sensor 5 mounted on the exhaust pipe 2 upstream of thecatalytic converter 3 (or more specifically at a position where exhaustgases from the cylinders of the internal combustion engine 1 are puttogether), an O₂ sensor (oxygen concentration sensor) 6 mounted as anexhaust gas sensor on the exhaust pipe 2 downstream of the catalyticconverter 3 (upstream of the catalytic converter 4), and a control unit7 for carrying out a control process (described later on) based onoutputs (detected values) from the sensors 5, 6. The control unit 7 issupplied with outputs from various sensors (not shown) for detectingoperating conditions of the internal combustion engine 1, including aengine speed sensor, an intake pressure sensor, a coolant temperaturesensor, etc.

The O₂ sensor 6 comprises an ordinary O₂ sensor for generating an outputVO2/OUT having a level depending on the oxygen concentration in theexhaust gas that has passed through the catalytic converter 3 (an outputrepresenting a detected value of the oxygen concentration of the exhaustgas). The oxygen concentration in the exhaust gas is commensurate withthe air-fuel ratio of an air-fuel mixture which, when combusted,produces the exhaust gas. The output VO2/OUT from the O₂ sensor 6 willchange with high sensitivity substantially linearly in proportion to theoxygen concentration in the exhaust gas, with the air-fuel ratiocorresponding to the oxygen concentration in the exhaust gas being in arelatively narrow range Δ close to a stoichiometric air-fuel ratio, asindicated by the solid-line curve a in FIG. 2. At oxygen concentrationscorresponding to air-fuel ratios outside of the range Δ, the outputVO2/OUT from the O₂ sensor 6 is saturated and is of a substantiallyconstant level.

The air-fuel ratio sensor 5 generates an out-put KACT representing adetected value of the air-fuel ratio of the exhaust gas that enters thecatalytic converter 3 (more specifically, an air-fuel ratio which isrecognized from the concentration of oxygen in the exhaust gas thatenters the catalytic converter 3). The air-fuel ratio sensor 5 comprisesa wide-range air-fuel ration sensor disclosed in Japanese laid-openpatent publication No. 4-369471 or U.S. Pat. No. 5,391,282 by theapplicant of the present application. As indicated by the solid-linecurve b in FIG. 2, the air-fuel ratio sensor 5 generates an output KACTwhose level is proportional to the concentration of oxygen in theexhaust gas in a wider range than the O₂ sensor 6. In the descriptionwhich follows, the air-fuel ratio sensor 5 will be referred to as “LAFsensor 5”, and the air-fuel ratio of the exhaust gas that enters thecatalytic converter 3 as “upstream-of-catalyst air-fuel ratio”.

The control unit 7 comprises a microcomputer including a CPU, a RAM, anda ROM (not shown), and has an exhaust-side control unit 7 a forperforming, in predetermined control cycles, a process of sequentiallygenerating a target air-fuel ratio KCMD for the upstream-of-catalystair-fuel ratio (which is also a target value for the output KACT of theLAF sensor 5) as a manipulated variable for determining theupstream-of-catalyst air-fuel ratio, and an engine-side control unit 7 bfor sequentially carryout out, in predetermined control cycles, aprocess of manipulating the upstream-of-catalyst air-fuel ratio byadjusting an amount of fuel supplied to the internal combustion engine 1depending on the target air-fuel ratio KCMD. These control units 7 a, 7b correspond respectively to a manipulated variable generating means andan air-fuel ratio manipulating means according to the present invention.The control unit 7 has a program stored in advance in the ROM forenabling the CPU to perform the control processes of the exhaust-sidecontrol unit 7 a and the engine-side control unit 7 b as described lateron. The control unit 7 has the ROM as a recording medium according tothe present invention.

In the present embodiments, the control cycles in which the controlunits 7 a, 7 b perform their respective processing sequences aredifferent from each other. Specifically, the control cycles of theprocessing sequence of the exhaust-side control unit 7 a have apredetermined fixed period (e.g., ranging from 30 to 100 ms) in view ofthe relatively long dead time present in an exhaust system E (describedlater on) including the catalytic converter 3, calculating loads, etc.The control cycles of the processing sequence of the engine-side controlunit 7 b have a period in synchronism with the crankshaft angle period(so-called TDC) of the internal combustion engine 1 because the processof adjusting the amount of fuel supplied to the internal combustionengine 1 needs to be in synchronism with combustion cycles of theinternal combustion engine 1. The period of the control cycles of theexhaust-side control unit 7 a is longer than the crankshaft angle period(TDC) of the internal combustion engine 1.

The processing sequences of the control units 7 a, 7 b will be describedbelow. The engine-side control unit 7 b has, as its functions, a basicfuel injection quantity calculator 8 for determining a basic fuelinjection quantity Tim to be injected into the internal combustionengine 1, a first correction coefficient calculator 9 for determining afirst correction coefficient KTOTAL to correct the basic fuel injectionquantity Tim, and a second correction coefficient calculator 10 fordetermining a second correction coefficient KCMDM to correct the basicfuel injection quantity Tim.

The basic fuel injection quantity calculator 8 determines a referencefuel injection quantity (an amount of supplied fuel) from the rotationalspeed NE and intake pressure PB of the internal combustion engine 1using a predetermined map, and corrects the determined reference fuelinjection quantity depending on the effective opening area of a throttlevalve (not shown) of the internal combustion engine 1, therebycalculating a basic fuel injection quantity Tim.

The first correction coefficient KTOTAL determined by the firstcorrection coefficient calculator 9 serves to correct the basic fuelinjection quantity Tim in view of an exhaust gas recirculation ratio ofthe internal combustion engine 1 (the proportion of an exhaust gascontained in an air-fuel mixture introduced into the internal combustionengine 10), an amount of purged fuel supplied to the internal combustionengine 1 when a canister (not shown) is purged, a coolant temperature,an intake temperature, etc. of the internal combustion engine 1.

The second correction coefficient KCMDM determined by the secondcorrection coefficient calculator 10 serves to correct the basic fuelinjection quantity Tim in view of the charging efficiency of an air-fuelmixture due to the cooling effect of fuel flowing into the internalcombustion engine 1 depending on a target air-fuel ratio KCMD which isdetermined by the exhaust-side control unit 7 a, as described later on.

The engine-side control unit 7 b corrects the basic fuel injectionquantity Tim with the first correction coefficient KTOTAL and the secondcorrection coefficient KCMDM by multiplying the basic fuel injectionquantity Tim by the first correction coefficient KTOTAL and the secondcorrection coefficient KCMDM, thus producing a demand fuel injectionquantity Tcyl for the internal combustion engine 1.

Specific details of processes for calculating the basic fuel injectionquantity Tim, the first correction coefficient KTOTAL, and the secondcorrection coefficient KCMDM are disclosed in detail in Japaneselaid-open patent publication No. 5-79374 and U.S. Pat. No. 5,253,630 bythe applicant of the present application, and will not be describedbelow. The engine-side control unit 7 b also has, in addition to theabove functions, a feedback controller 14 for feedback-controlling theair-fuel ratio of the internal combustion engine 1 by adjusting a fuelinjection quantity of the internal combustion engine 1 so as to convergethe output KACT of the LAF sensor 5 (the detected value of theupstream-of-catalyst air-fuel ratio) to the target air-fuel ratio KCMDwhich is sequentially calculated by the exhaust-side control unit 7 a(to be described in detail later).

The feedback controller 14 comprises a general feedback controller 15for feedback-controlling a total air-fuel ratio of the cylinders of theinternal combustion engine 1 and a local feedback controller 16 forfeedback-controlling an air-fuel ratio of each of the cylinders of theinternal combustion engine 1.

The general feedback controller 15 sequentially determines a feedbackcorrection coefficient KFB to correct the demand fuel injection quantityTcyl (by multiplying the demand fuel injection quantity Tcyl) so as toconverge the output KACT from the LAF sensor 5 to the target air-fuelratio KCMD. The general feedback controller 15 comprises a PIDcontroller 17 for generating a feedback manipulated variable KLAF as thefeedback correction coefficient KFB depending on the difference betweenthe output KACT from the LAF sensor 5 and the target air-fuel ratio KCMDaccording to a known PID control process, and an adaptive controller 18(indicated by “STR” in FIG. 1) for adaptively determining a feedbackmanipulated variable KSTR for determining the feedback correctioncoefficient KFB in view of changes in operating state of the internalcombustion engine 1 or characteristic changes thereof from the outputKACT from the LAF sensor 5 and the target air-fuel ratio KCMD.

In the present embodiment, the feedback manipulated variable KLAFgenerated by the PID controller 17 is of “1” and can be used directly asthe feedback correction coefficient KFB when the output KACT (thedetected value of the upstream-of-catalyst air-fuel ratio) from the LAFsensor 5 coincides with the target air-fuel ratio KCMD. The feedbackmanipulated variable KSTR generated by the adaptive controller 18becomes the target air-fuel ratio KCMD when the output KACT from the LAFsensor 5 is equal to the target air-fuel ratio KCMD. A feedbackmanipulated variable kstr (=KSTR/KCMD) which is produced by dividing thefeedback manipulated variable KSTR by the target air-fuel ratio KCMDwith a divider 19 can be used as the feedback correction coefficientKFB.

The feedback manipulated variable KLAF generated by the PID controller17 and the feedback manipulated variable kstr which is produced bydividing the feedback manipulated variable KSTR from the adaptivecontroller 18 by the target air-fuel ratio KCMD are selected one at atime by a switcher 20. A selected one of the feedback manipulatedvariable KLAF and the feedback manipulated variable KSTR is used as thefeedback correction coefficient KFB. The demand fuel injection quantityTcyl is corrected by being multiplied by the feedback correctioncoefficient KFB. Details of the general feedback controller 15(particularly, the adaptive controller 18) will be described later on.

The local feedback controller 16 comprises an observer 21 for estimatingreal air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respective cylindersfrom the output KACT from the LAF sensor 5, and a plurality of PIDcontrollers 22 (as many as the number of the cylinders) for determiningrespective feedback correction coefficients #nKLAF for fuel injectionquantities for the cylinders from the respective real air-fuel ratios#nA/F estimated by the observer 21 according to a PID control process soas to eliminate variations of the air-fuel ratios of the cylinders.

Briefly stated, the observer 21 estimates a real air-fuel ratio #nA/F ofeach of the cylinders as follows: A system from the internal combustionengine 1 to the LAF sensor 5 (where the exhaust gases from the cylindersare combined) is considered to be a system for generating an air-fuelratio detected by the LAF sensor 5 from a real air-fuel ratio #nA/F ofeach of the cylinders, and is modeled in view of a detection responsedelay (e.g., a time lag of first order) of the LAF sensor 5 and achronological contribution of the air-fuel ratio of each of thecylinders to the upstream-of-catalyst air-fuel ratio. Based on themodeled system, a real air-fuel ratio #nA/F of each of the cylinders isestimated from the output KACT from the LAF sensor 5.

Details of the observer 21 are disclosed in Japanese laid-open patentpublication No. 7-83094 or U.S. Pat. No. 5,531,208 by the applicant ofthe present application, and will not be described below.

Each of the PID controllers 22 of the local feedback controller 16divides the output signal KACT from the LAF sensor 5 by an average valueof the feedback correction coefficients #nKLAF determined by therespective PID controllers 22 in a preceding control cycle to produce aquotient value, and uses the quotient value as a target air-fuel ratiofor the corresponding cylinder. Each of the PID controllers 22 thendetermines a feedback correction coefficient #nKLAF in a present controlcycle so as to eliminate any difference between the target air-fuelratio and the corresponding real air-fuel ratio #nA/F determined by theobserver 21.

The local feedback controller 16 multiplies a value, which has beenproduced by multiplying the demand fuel injection quantity Tcyl by theselected feedback correction coefficient KFB produced by the generalfeedback controller 15, by the feedback correction coefficient #nKLAFfor each of the cylinders, thereby determining an output fuel injectionquantity #nTout (n=1, 2, 3, 4) for each of the cylinders.

The output fuel injection quantity #nTout thus determined for each ofthe cylinders is corrected for accumulated fuel particles on intake pipewalls of the internal combustion engine 1 by a fuel accumulationcorrector 23 in the engine-side control unit 7 b. The corrected outputfuel injection quantity #nTout is applied to each of fuel injectors (notshown) of the internal combustion engine 1, which injects fuel into eachof the cylinders with the corrected output fuel injection quantity#nTout.

The correction of the output fuel injection quantity in view ofaccumulated fuel particles on intake pipe walls is disclosed in detailin Japanese laid-open patent publication No. 8-21273 or U.S. Pat. No.5,568,799 by the applicant of the present application, and will not bedescribed in detail below. A sensor output selector 24 shown in FIG. 1serves to select the output KACT from the LAF sensor 5, which issuitable for the estimation of a real air-fuel ratio #nA/F of eachcylinder with the observer 21, depending on the operating state of theinternal combustion engine 1. Details of the sensor output selector 24are disclosed in detail in Japanese laid-open patent publication No.7-259588 or U.S. Pat. No. 5,540,209 by the applicant of the presentapplication, and will not be described in detail below.

The exhaust-side control unit 7 a has a subtractor 11 for sequentiallydetermining a difference kact (=KACT−FLAF/BASE) between the output KACTfrom the LAF sensor 5 and a predetermined air-fuel ratio reference valueFLAF/BASE and a subtractor 12 for sequentially determining a differenceVO2 (=VO2/OUT−VO2/TARGET) between the output VO2/OUT from the O₂ sensor6 and a target value VO2/TARGET therefor.

The target value VO2/TARGET for the output VO2/OUT from the O₂ sensor 6is a predetermined value as an output value of the O₂ sensor 6 in orderto achieve an optimum purifying capability of the catalytic converter 3(specifically, purification ratios for NOx, HC, CO, etc. in the exhaustgas), and is an output value that can be generated by the O₂ sensor 6 ina situation where the air-fuel ratio of the exhaust gas is present inthe range a close to a stoichiometric air-fuel ratio as shown in FIG. 2.In the present embodiment, the reference value FLAF/BASE with respect tothe output KACT from the LAF sensor 5 is set to a “stoichiometricair-fuel ratio” (constant value).

In the description which follows, the differences kact, VO2 determinedrespectively by the subtractors 11, 12 are referred to as a differentialoutput kact of the LAF sensor 5 and a differential output VO2 of the O₂sensor 6, respectively.

The exhaust-side control unit 7 a also has a target air-fuel ratiogeneration processor 13 for sequentially calculating the target air-fuelratio KCMD (the target value for the upstream-of-catalyst air-fuelratio) based on the data of the differential outputs kact, VO2 usedrespectively as the data of the output from the LAF sensor 5 and theoutput of the O₂ sensor 6.

The target air-fuel ratio generation processor 13 serves to control, asan object control system, an exhaust system (denoted by E in FIG. 1)including the catalytic converter 3, which ranges from the LAF sensor 5to the O₂ sensor 6 along the exhaust pipe 2. The target air-fuel ratiogeneration processor 13 sequentially determines the target air-fuelratio KCMD for the internal combustion engine 1 so as to converge(settle) the output VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET therefor according to a sliding mode control process(specifically an adaptive sliding mode control process) in view of adead time present in the exhaust system E, a dead time present in anair-fuel ratio manipulating system comprising the internal combustionengine 1 and the engine-side control unit 7 b, and behavioral changes ofthe exhaust system E.

In order to carry out the control process of the target air-fuel ratiogeneration processor 13, according to present embodiment, the exhaustsystem E is regarded as a system for generating the output VO2/OUT ofthe O₂ sensor 6 from the output KACT of the LAF sensor 5 (theupstream-of-catalyst air-fuel ratio detected by the LAF sensor 5) via adead time element and a response delay element, and a model isconstructed for expressing the behavior of the exhaust system E. Theair-fuel ratio manipulating system comprising the internal combustionengine 1 and the engine-side control unit 7 b is regarded as a systemfor generating the output KACT of the LAF sensor 5 from the targetair-fuel ratio KCMD via a dead time element, and a model is constructedfor expressing the behavior of the air-fuel ratio manipulating system.

With respect to the model of the exhaust system E (hereinafter referredto as “exhaust system model”), the behavior of the exhaust system E isexpressed by an autoregressive model of a discrete time system accordingto the equation (1) shown below (specifically, an autoregressive modelhaving a dead time in the differential output kact as the input quantityof the exhaust system E), using the differential output kact(=KACT−FLAF/BASE) from the LAF sensor 5 as the input quantity of theexhaust system E and the differential output VO2 (=VO2/OUT−VO2/TARGET)from the O₂ sensor 6 as the output quantity of the exhaust system E,instead of the output KACT of the LAF sensor 5 and the output VO2/OUT ofthe O₂ sensor 6.VO 2(k+1)=a 1·VO 2(k)+a 2·VO 2(k−1)+b 1·kact(k−d 1)  (1)

In the equation (1), “k” represents the ordinal number of adiscrete-time control cycle of the exhaust-side control unit 7 a, and“d1” the dead time of the exhaust system E (more specifically, the deadtime required until the upstream-of-catalyst air-fuel ratio detected ateach point of time by the LAF sensor 5 is reflected in the outputVO2/OUT of the O₂ sensor 6) as represented by the number of controlcycles. The actual dead time of the exhaust system E is closely relatedto the flow rate of the exhaust gas supplied to the catalytic converter3, and is basically longer as the flow rate of the exhaust gas issmaller. This is because as the flow rate of the exhaust gas is smaller,the time required for the exhaust gas to pass through the catalyticconverter 3 is longer. In the present embodiment, the flow rate of theexhaust gas supplied to the catalytic converter 3 is sequentiallygrasped, and the value of the dead time d1 in the exhaust system modelaccording to the equation (1) is variably set (the set value of the deadtime d1 will hereinafter be referred to as “set dead time d1”).

The first and second terms of the right side of the equation (1)correspond to a response delay element of the exhaust system E, thefirst term being a primary autoregressive term and the second term beinga secondary autoregressive term. In the first and second terms, “a1”,“a2” represent respective gain coefficients of the primaryautoregressive term and the secondary autoregressive term. Statedotherwise, these gain coefficients a1, a2 are relative to thedifferential output VO2 of the O₂ sensor 6 as an output quantity of theexhaust system E.

The third term of the right side of the equation (1) corresponds to adead time element of the exhaust system E, and represents thedifferential output kact of the LAF sensor 5 as an input quantity of theobject exhaust system E, including the dead time d1 of the exhaustsystem E. In the third term, “b1” represents a gain coefficient relativeto the dead time element (an input quantity having the dead time d1).

These gain coefficients “a1”, “a2”, “b1” are parameters to be set tocertain values for defining the behavior of the model of the exhaustsystem E, and are sequentially identified by an identifier which will bedescribed later on according to the present embodiment.

The exhaust system model expressed by the equation (1) thus expressesthe differential output VO2(k+1) of the O₂ sensor as the input quantityof the exhaust system E in each control cycle of the exhaust-sidecontrol unit 7 a, with the differential outputs VO2(k), VO2(k−1) in pastcontrol cycles prior to that control cycle and the differential outputkact(k−d1) of the LAF sensor 5 as the input quantity(upstream-of-catalyst air-fuel ratio) of the exhaust system E in acontrol cycle prior to the dead time d1 of the exhaust system E.

With respect to the model of the air-fuel ratio manipulating systemcomprising the internal combustion engine 1 and the engine-side controlunit 7 b (hereinafter referred to as “air-fuel ratio manipulating systemmodel”), the difference kcmd (=KCMD−FLAF/BASE, hereinafter referred toas “target differential air-fuel ratio kcmd”) between the targetair-fuel ratio KCMD and the air-fuel ratio reference value FLAF/BASE isregarded as an input quantity of the air-fuel ratio manipulating system,the differential output kact of the LAF sensor 5 as an output quantityof the air-fuel ratio manipulating system, and the behavior of theair-fuel ratio manipulating system model is expressed by a modelaccording to the following equation (2):kact(k)=kcmd(k−d 2)  (2)

In the equation (2), “d2” represents the dead time of the air-fuel ratiomanipulating system (more specifically, the dead time required until thetarget air-fuel ratio KCMD at each point of time is reflected in theoutput KACT of the LAF sensor 5) in terms of the number of controlcycles of the exhaust-side control unit 7 a. The actual dead time of theair-fuel ratio manipulating system is closely related to the flow rateof the exhaust gas supplied to the catalytic converter 3, as with thedead time of the exhaust system E, and is basically loner as the flowrate of the exhaust gas is smaller. This because as the flow rate of theexhaust gas is smaller, the rotational speed of the internal combustionengine 1 is lower (the crankshaft angle period is longer), and theperiod of the control cycles of the engine-side control unit 7 b of theair-fuel ratio manipulating system is longer. In the present embodiment,therefore, the flow rate of the exhaust gas supplied to the catalyticconverter 3 is sequentially recognized, and the value of the dead timet2 in the air-fuel ratio manipulating system according to the equation(2) is variably set (the set value of the dead time d2 will hereinafterbe referred to as “set dead time d2”).

The air-fuel ratio manipulating system model expressed by the equation(2) regards the air-fuel ratio manipulating system as a system whereinthe differential output kact of the LAF sensor 5 as the output quantity(upstream-of-catalyst air-fuel ratio) of the air-fuel ratio manipulatingsystem coincides with the target differential air-fuel ratio kcmd as theinput quantity of the air-fuel ratio manipulating system at a time priorto the dead time t2 in the air-fuel ratio manipulating system, andexpresses the behavior of the air-fuel ratio manipulating system.

The air-fuel ratio manipulating system actually includes a responsedelay element caused by the internal combustion engine 1, other than adead time element. Since a response delay of the upstream-of-catalystair-fuel ratio with respect to the target air-fuel ratio KCMD isbasically compensated for by the feedback controller 14 (particularlythe adaptive controller 18) of the engine-side control unit 7 b, therewill arise no problem if a response delay element caused by the internalcombustion engine 1 is not taken into account in the air-fuel ratiomanipulating system as viewed from the exhaust-side control unit 7 a.

The target air-fuel ratio generation processor 13 according to thepresent invention carries out the process for sequentially calculatingthe target air-fuel ratio KCMD according to an algorithm that isconstructed based on the exhaust system model expressed by the equation(1) and the air-fuel ratio manipulating system model expressed by theequation (2) in control cycles of the exhaust-side control unit 7 a. Inorder to carry out the above process, the target air-fuel ratiogeneration processor 13 has its functions as shown in FIG. 3.

The target air-fuel ratio generation processor 13 comprises a flow ratedata generating means 28 for sequentially calculating an estimated valueABSV of the flow rate of the exhaust gas supplied to the catalyticconverter 3 (hereinafter referred to as “estimated exhaust gas volumeABSV”) from the detected values of the rotational speed NE and theintake pressure PB of the internal combustion engine 1, and a dead timesetting means 29 for sequentially setting the set dead times d1, d2 ofthe exhaust system model and the air-fuel ratio manipulating systemmodel, respectively, depending on the estimated exhaust gas volume ABSV.

Since the flow rate of the exhaust gas supplied to the catalyticconverter 3 is proportional to the product of the rotational speed NEand the intake pressure PB of the internal combustion engine 1, the deadtime setting means 29 sequentially calculates the estimated exhaust gasvolume ABSV from the detected values (present values) of the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1according to the following equation (3): $\begin{matrix}{{ABSV} = {\frac{NE}{1500} \cdot {PB} \cdot {SVPRA}}} & (3)\end{matrix}$

In the equation (3), SVPRA represents a predetermined constant dependingon the displacement (cylinder volume) of the internal combustion engine1. In the present embodiment, the flow rate of the exhaust gas when therotational speed NE of the internal combustion engine 1 is 1500 rpm isused as a reference. Therefore, the rotational speed NE is divided by“1500” in the above equation (3).

The dead time setting means 29 sequentially determines the set dead timed1 as a value representing the actual dead time of the exhaust system Efrom the value of the estimated gas volume ABSV sequentially calculatedby the flow rate data generating means 28 according to a data table thatis preset as indicated by the solid-line curve c in FIG. 4, for example.Similarly, the dead time setting means 29 sequentially determines theset dead time d2 as a value representing the actual dead time of theair-fuel ratio manipulating system from the value of the estimated gasvolume ABSV according to a data table that is preset as indicated by thesolid-line curve d in FIG. 4.

The above data tables are established based on experimentation orsimulation. Since the actual dead time of the exhaust system E isbasically longer as the flow rate of the exhaust gas supplied to thecatalytic converter 3 is smaller, as described above, the set dead timed1 represented by the solid-line curve c in FIG. 4 varies according tosuch a tendency with respect to the estimated gas volume ABSV. Likewise,since the actual dead time of the air-fuel ratio manipulating system isbasically longer as the flow rate of the exhaust gas supplied to thecatalytic converter 3 is smaller, the set dead time d2 represented bythe solid-line curve d in FIG. 4 varies according to such a tendencywith respect to the estimated gas volume ABSV. Moreover, inasmuch as thedegree of changes of the actual dead time of the air-fuel ratiomanipulating system with respect to the flow rate of the exhaust gas issmaller than the degree of changes of the actual dead time of theexhaust system E, the degree of changes of the set dead time d2 withrespect to the estimated gas volume ABSV is smaller than the degree ofchanges of the set dead time d1 in the data table shown in FIG. 4.

In the data table shown in FIG. 4, the set dead times d1, d2continuously change with respect to the estimated gas volume ABSV. Sincethe set dead times d1, d2 in the exhaust system model and the air-fuelratio manipulating system model are expressed in terms of the number ofcontrol cycles of the exhaust-side control unit 7 a, the set dead timesd1, d2 need to be of integral values. Therefore, the dead time settingmeans 29 actually determines, as set dead times d1, d2, values that areproduced by rounding off the fractions of the values of the set deadtimes d1, d2 that are determined based on the data table shown in FIG.4, for example.

In the present embodiment, the flow rate of the exhaust gas supplied tothe catalytic converter 3 is estimated from the rotational speed NE andthe intake pressure PB of the internal combustion engine 1. However, theflow rate of the exhaust gas may be directly determined using a flowsensor or the like.

The target air-fuel ratio generation processor 13 comprises anidentifier (identifying means) 25 for sequentially identifying values ofthe gain coefficients a1, a2, b1 that are parameters for the exhaustsystem model, an estimator (estimating means) 26 for sequentiallydetermining in each control cycle an estimated value VO2 bar of thedifferential output VO2 from the O₂ sensor 6 (hereinafter referred to as“estimated differential output VO2 bar”) after the total set dead time d(=d1+d2) which is the sum of the set dead time d1 of the exhaust systemE and the set dead time d2 of the air-fuel ratio manipulating system,and a sliding mode controller 27 for sequentially determining the targetair-fuel ratio KCMD according to an adaptive sliding mode controlprocess.

The algorithm of a processing operation to be carried out by theidentifier 25, the estimator 26, and the sliding mode controller 27 isconstructed based on the exhaust system model and the air-fuel ratiomanipulating system model, as follows:

With respect to the identifier 25, the gain coefficients of the actualexhaust system E which correspond to the gain coefficients a1, a2, b1 ofthe exhaust system model generally change depending on the behavior ofthe exhaust system E and chronological characteristic changes of theexhaust system E. Therefore, in order to minimize a modeling error ofthe exhaust system model (the equation (1)) with respect to the actualexhaust system E for increasing the accuracy of the model, it ispreferable to identify the gain coefficients a1, a2, b1 in real-timesuitably depending on the actual behavior of the exhaust system E.

The identifier 25 serves to identify the gain coefficients a1, a2, b1sequentially on a real-time basis for the purpose of minimizing amodeling error of the exhaust system model. The identifier 25 carriesout its identifying process as follows:

In each control cycle of the exhaust-side control unit 7 a, theidentifier 25 determines an identified value VO2(k) hat of thedifferential output VO2 from the O₂ sensor 6 (hereinafter referred to as“identified differential output VO2(k) hat”) on the exhaust systemmodel, using the identified gain coefficients a1 hat, a2 hat, b1 hat ofthe presently established exhaust system model, i.e., identified gaincoefficients a1(k−1) hat, a2(k−1) hat, b1(k−1) hat determined in apreceding control cycle, past data of the differential output kact fromthe LAF sensor 5 and the differential output VO2 from the O₂ sensor 6,and the latest value of the set dead time d1 of the exhaust system Ethat has been set by the dead time setting means 29, according to thefollowing equation (4):VÔ 2(k)=a^1(k−1)·VO 2(k−1)+a^2(k−1)·VO 2(k−2)+b^1(k−1)·kact(k−d1−1)  (4)

The equation (4) corresponds to the equation (1) which is shifted intothe past by one control cycle with the gain coefficients a1, a2, b1being replaced with the respective identified gain coefficients a1(k−1)hat, a2(k−1) hat, b1(k−1) hat, and the latest value of the set dead timed1 used as the dead time d1 of the exhaust system E.

If vectors Θ, ζ defined by the following equations (5), (6) areintroduced (the letter T in the equations (5), (6) represents atransposition), then the equation (4) is expressed by the equation (7):Θ^(T)(k)=[a^1(k)a^2(k)b^1(k)]  (5)ζ^(T)(k)=[VO 2(k−1)VO 2(k−2)kact(k−d 1−1)]  (6)VÔ 2(k)=Θ^(T)(k−1)·ζ(k)  (7)

The identifier 25 also determines a difference id/e(k) between theidentified differential output VO2(k) hat from the O₂ sensor 6 which isdetermined by the equation (4) or (7) and the present differentialoutput VO2(k) from the O₂ sensor 6, as representing a modeling error ofthe exhaust system model with respect to the actual exhaust system E(hereinafter the difference id/e will be referred to as “identifiederror id/e”), according to the following equation (8):id/e(k)=VO 2(k)−VÔ 2(k)  (8)

The identifier 25 further determines new identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat, stated otherwise, a new vector Θ(k)having these identified gain coefficients as elements (hereinafter thenew vector Θ(k) will be referred to as “identified gain coefficientvector Θ”), in order to minimize the identified error id/e, according tothe equation (9) given below. That is, the identifier 25 varies theidentified gain coefficients a1 hat (k−1), a2 hat (k−1), b1 hat (k−1)determined in the preceding control cycle by a quantity proportional tothe identified error id/e for thereby determining the new identifiedgain coefficients a1(k) hat, a2(k) hat, b1(k) hat.Θ(k)=Θ(k−1)+Kθ(k)·id/e(k)  (9)where Kθ represents a cubic vector determined by the following equation(10) (a gain coefficient vector for determining a change depending onthe identified error id/e of each of the identified gain coefficients a1hat, a2 hat, b1 hat): $\begin{matrix}{{K\quad\theta\quad(k)} = \frac{{P\left( {k - 1} \right)} \cdot {\xi(k)}}{1 + {{{\xi^{T}(k)} \cdot {P\left( {k - 1} \right)} \cdot \xi}\quad(k)}}} & (10)\end{matrix}$where P represents a cubic square matrix determined by a recursiveformula expressed by the following equation (11): $\begin{matrix}{{P(k)} = {{\frac{1}{\lambda_{1}(k)} \cdot \left\lbrack {I - \frac{{\lambda_{2}(k)} \cdot {P\left( {k - 1} \right)} \cdot {\xi(k)} \cdot {\xi^{T}(k)}}{{\lambda_{2}(k)} + {{{\lambda_{2}(k)} \cdot {\xi^{T}(k)} \cdot P}{\left( {k - 1} \right) \cdot \xi}(k)}}} \right\rbrack \cdot P}\quad\left( {K - 1} \right)}} & (11)\end{matrix}$where I represents a unit matrix.

In the equation (11), λ₁, λ₂ are established to satisfy the conditions0<λ₁≦1 and 0<λ₂<2, and an initial value P(0) of P represents a diagonalmatrix whose diagonal components are positive numbers.

Depending on how the values of λ₁, λ₂ in the equation (11) areestablished, any one of various specific algorithms including a fixedgain method, a degressive gain method, a method of weighted leastsquares, a method of least squares, a fixed tracing method, etc. may beemployed. According to the present embodiment, the algorithm of a methodof weighted least squares is employed, and the values of λ₁, λ₂ are that0<λ₁<1, λ₂=1.

“λ₁” represents a weighted parameter according to a method of weightedleast squares. In the present embodiment, the value of the weightedparameter λ₁ is variably set depending on the estimated exhaust gasvolume ABSV that is sequentially calculated by the flow rate datagenerating means 28 (as a result, depending on the set dead time d1).

Specifically, in the present embodiment, the identifier 25 sets, in eachcontrol cycle of the exhaust-side control unit 7 a, the value of theweighted parameter λ₁ from the latest value of the estimated exhaust gasvolume ABSV determined by the flow rate data generating means 28, basedon a predetermined data table shown in FIG. 5. In the data table shownin FIG. 5, the value of the weighted parameter λ₁ is greater,approaching “1”, as the estimated exhaust gas volume ABSV is smaller.The identifier 25 uses the value of the weighted parameter λ₁ thus setdepending on the estimated exhaust gas volume ABSV for updating thematrix P(k) according to the equation (11) in each control cycle.

Basically, the identifier 25 sequentially determines in each controlcycle the identified gain coefficients a1 hat, a2 hat, b1 hat of theexhaust system model according to the above algorithm (calculatingoperation), i.e., the algorithm of a sequential method of weighted leastsquares, in order to minimize the identified error id/e.

The calculating operation described above is the basic algorithm that iscarried out by the identifier 25. The identifier 25 performs additionalprocesses such as a limiting process, on the identified gaincoefficients a1 hat, a2 hat, b1 hat in order to determine them. Suchoperations of the identifier 25 will be described later on.

The estimator 26 sequentially determines in each control cycle theestimated differential output VO2 bar which is an estimated value of thedifferential output VO2 from the O₂ sensor 6 after the total set deadtime d (=d1+d2) in order to compensate for the effect of the dead timed1 of the exhaust system E and the effect of the dead time d2 of theair-fuel ratio manipulating system for the calculation of the targetair-fuel ratio KCMD with the sliding mode controller 27 as described indetail later on. The algorithm for the estimator 26 to determine theestimated differential output VO2 bar is constructed as follows:

If the equation (2) expressing the air-fuel ratio manipulating systemmodel is applied to the equation (1) expressing the exhaust systemmodel, then the equation (1) can be rewritten as the following equation(12): $\begin{matrix}{\quad\begin{matrix}{{{VO2}\left( {k + 1} \right)} = {{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} + {{b1} \cdot {{kcmd}\left( {k - {d1} - {d2}} \right)}}}} \\{= {{{a1} \cdot {{VO2}(k)}} + {{a2} \cdot {{VO2}\left( {k - 1} \right)}} + {{b1} \cdot {{kcmd}\left( {k - d} \right)}}}}\end{matrix}} & (12)\end{matrix}$

The equation (12) expresses the behavior of a system which is acombination of the exhaust system E and the air-fuel manipulating systemas a discrete time system, regarding such a system as a system forgenerating the differential output VO2 from the O₂ sensor 6 from thetarget differential air-fuel ratio kcmd via dead time elements of theexhaust system E and the air-fuel manipulating system and a responsedelay element of the exhaust system E.

By using the equation (12), the estimated differential output VO2(k+d)bar after the total set dead time d in each control cycle can beexpressed using time-series data VO2(k), VO2(k−1) of present and pastvalues of the differential output VO2 of the O₂ sensor 6 and time-seriesdata kcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd (=KCMD−FLAF/BASE) which corresponds tothe target air-fuel ratio KCMD determined by the sliding mode controller27 (its specific process of determining the target air-fuel ratio KCMDwill be described later on), according to the following equation (13):$\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{\alpha\quad{1 \cdot {{VO2}(k)}}} + {\alpha\quad{2 \cdot {{VO2}\left( {k - 1} \right)}}} + {\sum\limits_{j = 1}^{d}{\beta\quad{j \cdot {{kcmd}\left( {k - j} \right)}}}}}} & (13)\end{matrix}$where

-   -   α1=the first-row, first-column element of A^(d),    -   α2=the first-row, second-column element of A^(d),    -   βj=the first-row elements of A^(j−1)·B $A = \begin{bmatrix}        {a1} & {a2} \\        1 & 0        \end{bmatrix}$ $B = \begin{bmatrix}        {b1} \\        0        \end{bmatrix}$

In the equation (13), “α1”, “α2” represent the first-row, first-columnelement and the first-row, second-column element, respectively, of thepower A^(d) (d: total dead time) of the matrix A defined as describedabove with respect to the equation (13), and “βj” (j=1, 2, . . . , d)represents the first-row elements of the product A^(j−1)·B of the powerA^(j−1) (j=1, 2, . . . , d) of the matrix A and the vector B defined asdescribed above with respect to the equation (13).

Of the time-series data kcmd(k−j) (j=1, 2, . . . , d) of the past valuesof the target combined differential air-fuel ratio kcmd according to theequation (13), the time-series data kcmd(k−d2), kcmd(k−d2−1), . . . ,kcmd(k−d) from the present prior to the dead time d2 of the air-fuelmanipulating system can be replaced respectively with data kact(k),kact(k−1), . . . , kact(k−d+d2) obtained prior to the present time ofthe differential output kact of the LAF sensor 5 according the aboveequation (2). When the time-series data are thus replaced, the followingequation (14) is obtained: $\begin{matrix}\begin{matrix}{{\overset{\_}{VO2}\quad\left( {k + d} \right)} = {{\alpha\quad{1 \cdot {{VO2}(k)}}} + {\alpha\quad{2 \cdot {{VO2}\left( {k - 1} \right)}}} +}} \\{{\sum\limits_{j = 1}^{{d2} - 1}{\beta\quad{j \cdot {{kcmd}\left( {k - j} \right)}}}} +} \\{{\sum\limits_{i = 0}^{d - {d2}}{\beta\quad i}} + {{d2} \cdot {{kact}\left( {k - i} \right)}}} \\{= {{\alpha\quad{1 \cdot {{VO2}(k)}}} + {\alpha\quad{2 \cdot {{VO2}\left( {k - 1} \right)}}} +}} \\{{\sum\limits_{j = 1}^{{d2} - 1}{\beta\quad{j \cdot {{kcmd}\left( {k - j} \right)}}}} +} \\{{\sum\limits_{i = 0}^{d1}{\beta\quad i}} + {{d2} \cdot {{kact}\left( {k - 1} \right)}}}\end{matrix} & (14)\end{matrix}$

The equation (14) is a basic formula for the estimator 26 tosequentially determine the estimated differential output VO2(k+d) bar.Stated otherwise, the estimator 26 determines, in each control cycle,the estimated differential output VO2(k+d) bar of the O₂ sensor 6according to the equation (14), using the time-series data VO2(k),VO2(k−1) of the present and past values of the differential output VO2of the O₂ sensor 6, the time-series data kcmd(k−j) (j=1, . . . , d2−1)of the past values of the target differential air-fuel ratio kcmd whichrepresents the target air-fuel ratio KCMD determined in the past by thesliding mode controller 27, and the time-series data kact(k−i) (i=0, . .. , d1) of the present and past values of the differential output kactof the LAF sensor 5.

The values of the coefficients α1, α2, βj (j=1, 2, . . . , d) requiredto calculate the estimated differential output VO2(k+d) bar according tothe equation (14) basically employ the identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat which are the latest identified valuesof the gain coefficients a1, a2, b1 (which are elements of the vectorsA, B defined with respect to the equation (13)). The values of the deadtimes d1, d2 required in the equation (14) comprise the latest values ofthe set dead times d1, d2 that are set by the dead time setting means 29as described above.

In the present embodiment, the set dead times d1, d2 used in theequation (14) change depending on the estimated exhaust gas volume ABSV,and the number of data of the target differential air-fuel ratio kcmdand data of the differential output kact of the LAF sensor 5 which arerequired to calculate the estimated differential output VO(k+d) baraccording to the equation (14) also changes depending on the set deadtimes d1, d2. In this case, the set dead time d2 of the air-fuel ratiomanipulating system may become “1” (in the present embodiment d1>d2≧1,see FIG. 4). If the dead time d2 of the air-fuel ratio manipulatingsystem becomes “1”, then all the time-series data kcmd(k−j) (j=1, 2, . .. , d) of the past values of the target differential air-fuel ratio kcmdin the equation (13) may be replaced with the time-series data kact(k),kact(k−1), . . . , kact(k−d+d2), respectively, prior to the presenttime, of the differential output kact of the LAF sensor 5. In this case,the equation (13) is rewritten into the following equation (15) whichdoes not include the data of the target differential air-fuel ratiokcmd: $\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{\alpha\quad{1 \cdot {{VO2}(k)}}} + {\alpha\quad{2 \cdot {{VO2}\left( {k - 1} \right)}}} + {\sum\limits_{j = 1}^{d - 1}{\beta\quad j}} + {1 \cdot {{kact}\left( {k - j} \right)}}}} & (15)\end{matrix}$

Specifically, if the value of the set dead time d2 is “1”, then theestimated differential output VO2(k+d) bar of the O₂ sensor 6 can bedetermined using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6, the time-series datakact(k−j) (j=0, 1, . . . , d−1) of the present and past values of thedifferential output kact of the LAF sensor 5, the coefficients α1, α2,βj (j=1, 2, . . . , d) determined by the identified gain coefficients a1hat, a2 hat, b1 hat, and the total set dead time d (=d1+d2) which is thesum of the set dead times d1, d2.

In the present embodiment, therefore, if the set dead time d2 is d2>1,then the estimator 26 determines the estimated differential outputVO2(k+d) bar according to the equation (14), and if the set dead time d2is d2=1, then the estimator 26 determines the estimated differentialoutput VO2(k+d) bar according to the equation (15).

The estimated differential output VO2(k+d) bar may be determinedaccording to the equation (13) without using the data of thedifferential output kact of the LAF sensor 5. In this case, theestimated differential output VO2(k+d) bar of the O₂ sensor 6 isdetermined using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6, the time-series datakcmd(k−j) (j=1, 2, . . . , d) of the past values of the targetdifferential air-fuel ratio kcmd, the coefficients α1, α2, βj (j=1, 2, .. . , d) determined by the identified gain coefficients a1 hat, a2 hat,b1 hat, and the total set dead time d (=d1+d2) which is the sum of theset dead times d1, d2. It is also possible to determine the estimateddifferential output VO2(k+d) bar according to an equation where only aportion of the time-series data of the target differential air-fuelratio kcmd prior to the set dead time d2 in the equation (13) isreplaced with the differential output kact of the LAF sensor 5. However,for increasing the reliability of the estimated differential outputVO2(k+d) bar, it is preferable to determine the estimated differentialoutput VO2(k+d) bar according to the equation (14) or (15) which uses,as much as possible, the data of the differential output kact of the LAFsensor 5 that reflects the actual behavior of the internal combustionengine 1, etc.

The algorithm described above is a basic algorithm for the estimator 26to determine, in each control cycle, the estimated differential outputVO2(k+d) bar that is an estimated value after the total dead time d ofthe differential output VO2 of the O₂ sensor 6.

The sliding mode controller 27 will be described in detail below.

The sliding mode controller 27 sequentially calculates an input quantityto be given to the exhaust system E (which is specifically a targetvalue for the difference between the output KACT of the LAF sensor 5(the detected value of the air-fuel ratio) and the air-fuel ratioreference value FLAF/BASE, which is equal to the target differentialair-fuel ratio kcmd, the input quantity will be referred to as “SLDmanipulating input Usl”) in order to cause the output VO2/OUT of the O₂sensor 6 to converge to the target value VO2/TARGET (to converge thedifferential output VO2 of the O₂ sensor 6 to “0”) according to anadaptive sliding mode control process which incorporates an adaptivecontrol law (adaptive algorithm) for minimizing the effect of adisturbance, in a normal sliding mode control process, and sequentiallydetermines the target air-fuel ratio KCMD from the calculated SLDmanipulating input Usl. An algorithm for carrying out the adaptivesliding mode control process is constructed as follows:

A switching function required for the adaptive sliding mode controlprocess carried out by the sliding mode controller 27 and a hyperplanedefined by the switching function (also referred to as a slip plane)will first be described below.

According to a basic concept of the sliding mode control process in thepresent embodiment, the differential output VO2(k) of the O₂ sensor 6obtained in each control cycle and the differential output VO2(k−1)obtained in a preceding control cycle are used as a state quantity to becontrolled, and a switching function a for the sliding mode controlprocess is defined according to the equation (16) shown below.Specifically, the switching function σ is defined by a linear functionwhose components are represented by the time-series data VO2(k),VO2(k−1) of the differential output VO2 of the O₂ sensor 6.σ(k)=s 1·VO 2(k)+s 2·VO 2(k−1)=S·X  (16)where ${S = \left\lbrack \begin{matrix}{s1} & {s2}\end{matrix}\quad \right\rbrack},{X = \begin{bmatrix}{{VO2}(k)} \\{{VO2}\left( {k - 1} \right)}\end{bmatrix}}$

The coefficients s1, s2 relative to the respective components VO2(k),VO2(k−1) of the switching function a are set in order to meet thecondition of the following equation (17): $\begin{matrix}{{- 1} < \frac{s2}{s1} < 1} & (17)\end{matrix}$

In the present embodiment, for the sake of brevity, the coefficient s1is set to s1=1 (s2/s1=s2), and the value of the coefficient s2 isestablished to satisfy the condition: −1<s2<1.

With the switching function σ thus defined, the hyperplane for thesliding mode control process is defined by the equation σ=0. Since thestate quantity X is of the second degree, the hyperplane σ=0 isrepresented by a straight line as shown in FIG. 6. The hyperplane iscalled a switching line or a switching plane depending on the degree ofa topological space.

In the present embodiment, the time-series data of the estimateddifferential output VO2 bar determined by the estimator 26 is used as astate quantity representative of the variable components of theswitching function, as described later on.

The adaptive sliding mode control process used in the present embodimentserves to converge the state quantity X onto the hyperplane σ=0according to a reaching law which is a control law for converging thestate quantity X (=VO2(k), VO2(k−1)) onto the hyperplane σ=0 (forconverging the value of the switching function σ to “0”) and an adaptivelaw (adaptive algorithm) which is a control law for compensating for theeffect of a disturbance in converging the state quantity X onto thehyperplane σ=0 (mode 1 in FIG. 6). While holding the state quantity Xonto the hyperplane σ=0 according to a so-called equivalent controlinput, the state quantity X is converged to a balanced point on thehyperplane σ=0 where VO2(k)=VO2(k−1)=0, i.e., a point where time-seriesdata VO2/OUT(k), VO2/OUT(k−1) of the output VO2/OUT of the O₂ sensor 6are equal to the target value VO2/TARGET (mode 2 in FIG. 6).

The SLD manipulating input Usl (=the target differential air-fuel ratiokcmd) to be generated by the sliding mode controller 27 for convergingthe state quantity X toward the balanced point on the hyperplane σ=0 isexpressed as the sum of an equivalent control input Ueq to be applied tothe exhaust system E according to the control law for converging thestate quantity X onto the hyperplane σ=0, an input quantity componentUrch (hereinafter referred to as “reaching law input Urch”) to beapplied to the exhaust system E according to the reaching law, and aninput quantity component Uadp (hereinafter referred to as “adaptive lawinput Uadp”) to be applied to the exhaust system E according to theadaptive law, according to the following equation (18).Usl=Ueq+Urch+Uadp  (18)

In the present embodiment, the equivalent control input Ueq, thereaching law input Urch, and the adaptive law input Uadp are determinedon the basis of the above equation (12) where the exhaust system modeland the air-fuel ratio manipulating system model are combined, asfollows:

The equivalent control input Ueq which is an input quantity component tobe applied to the exhaust system E for holding the state quantity X onthe hyperplane σ=0 is the target differential air-fuel ratio kcmd whichsatisfies the condition: a(k+1)=σ(k)=0. Using the equations (12), (16),the equivalent control input Ueq which satisfies the above condition isgiven by the following equation (19): $\quad\begin{matrix}\begin{matrix}{{{Ueq}(k)} = {{- \left( {S \cdot B} \right)^{- 1}} \cdot \left\{ {S \cdot \left( {A - 1} \right)} \right\} \cdot {X\left( {k + d} \right)}}} \\{= {\frac{- 1}{s1b1} \cdot \left\{ {{{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot {VO2}}\left( {k + d} \right)} +} \right.}} \\\left. {\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot {{VO2}\left( {k + d - 1} \right)}} \right\}\end{matrix} & (19)\end{matrix}$

The equation (19) is a basic formula for determining the equivalentcontrol input Ueq(k) in each control cycle.

According to the present embodiment, the reaching law input Urch isbasically determined according to the following equation (20):$\quad\begin{matrix}\begin{matrix}{{{Urch}(k)} = {{- \left( {S \cdot B} \right)^{- 1}} \cdot F \cdot {\sigma\left( {k + d} \right)}}} \\{= {\frac{- 1}{s1b1} \cdot F \cdot {\sigma\left( {k + d} \right)}}}\end{matrix} & (20)\end{matrix}$

Specifically, the reaching law input Urch is determined in proportion tothe value σ(k+d) of the switching function a after the total dead timed, in view of the dead times of the exhaust system E and the air-fuelratio manipulating system.

The coefficient F in the equation (20) (which determines the gain of thereaching law) is established to satisfy the condition expressed by thefollowing equation (21):0<F<2  (21)

-   -   (preferably, 0<F<1)

The condition of the equation (21) is a condition for stably convergingthe value of the switching function a onto the hyperplane σ=0. Thepreferable condition in the equation (21) is a condition suitable forpreventing the value of the switching function a from oscillating(so-called chattering) with respect to the hyperplane σ=0.

In the present embodiment, the adaptive law input Uadp is basicallydetermined according to the following equation (22) (ΔT in the equation(22) represents the period of the control cycles of the exhaust-sidecontrol unit 7 a): $\quad\begin{matrix}\begin{matrix}{{{Uadp}(k)} = {{- \left( {S \cdot B} \right)^{- 1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma(i)} \cdot \Delta}\quad T} \right)}}} \\{= {\frac{- 1}{s1b1} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\sigma(i)} \cdot \Delta}\quad T} \right)}}}\end{matrix} & (22)\end{matrix}$

The adaptive law input Uadp is determined in proportion to an integratedvalue (which corresponds to an integral of the values of the switchingfunction σ) of the product of values of the switching function σ and theperiod ΔT of the control cycles of the exhaust-side control unit 7 auntil after the total dead time d, in view of the dead times of theexhaust system E and the air-fuel ratio manipulating system.

The coefficient G (which determines the gain of the adaptive law) in theequation (22) is established to satisfy the condition of the followingequation (23): $\begin{matrix}{{G = {J \cdot \frac{2 - F}{\Delta\quad T}}}\left( {0 < J < 2} \right)} & (23)\end{matrix}$

The condition of the equation (23) is a condition for converging thevalue of the switching function σ stably onto the hyperplane σ=0regardless of disturbances, etc.

A specific process of deriving conditions for establishing the equations(17), (21), (23) is described in detail in Japanese patent applicationNo. 11-93741 or U.S. Pat. No. 6,082,099 by the applicant of the presentapplication, and will not be described in detail below.

In the present embodiment, the sliding mode controller 27 determines thesum (Ueq+Urch+Uadp) of the equivalent control input Ueq, the reachinglaw input Urch, and the adaptive law input Uadp determined according tothe respective equations (19), (20), (22) as the SLD manipulating inputUsl to be applied to the exhaust system E. However, the differentialoutputs VO2(K+d), VO2(k+d−1) of the O₂ sensor 6 and the value σ(k+d) ofthe switching function σ, etc. used in the equations (19), (20), (22)cannot directly be obtained as they are values in the future.

According to the present embodiment, therefore, the sliding modecontroller 27 actually uses the estimated differential outputs VO2(k+d)bar, VO2(k+d−1) bar determined by the estimator 26, instead of thedifferential outputs VO2(K+d), VO2(k+d−1) from the O₂ sensor 6 fordetermining the equivalent control input Ueq according to the equation(19), and calculates the equivalent control input Ueq in each controlcycle according to the following equation (24): $\quad\begin{matrix}{{{Ueq}(k)} = {\frac{- 1}{s1b1}\quad\left\{ {{\left\lbrack {{{s1} \cdot \left( {{a1} - 1} \right)} + {s2}} \right\rbrack \cdot {\overset{\_}{VO2}\left( {k + d} \right)}} + {\left( {{{s1} \cdot {a2}} - {s2}} \right) \cdot {\overset{\_}{VO2}\left( {k + d - 1} \right)}}} \right\}}} & (24)\end{matrix}$

According to the present embodiment, furthermore, the sliding modecontroller 27 actually uses time-series data of the estimateddifferential output VO2 bar sequentially determined by the estimator 26as described above as a state quantity to be controlled, and defines aswitching function a bar according to the following equation (25) (theswitching function a bar corresponds to time-series data of thedifferential output VO2 in the equation (16) which is replaced withtime-series data of the estimated differential output VO2 bar), in placeof the switching function a established according to the equation (16):{overscore (σ(k))}=s 1·{overscore (VO 2)}(k)+s 2·{overscore (VO2)}(k−1)  (25)

The sliding mode controller 27 calculates the reaching law input Urch ineach control cycle according to the following equation (26), using theswitching function a bar represented by the equation (25), rather thanthe value of the switching function σ for determining the reaching lawinput Urch according to the equation (20): $\begin{matrix}{{{Urch}(k)} = {\frac{- 1}{{s1} \cdot {b1}} \cdot F \cdot {\overset{\_}{\sigma}\left( {k + d} \right)}}} & (26)\end{matrix}$

Similarly, the sliding mode controller 27 calculates the adaptive lawinput Uadp in each control cycle according to the following equation(27), using the value of the switching function a bar represented by theequation (25), rather than the value of the switching function σ fordetermining the adaptive law input Uadp according to the equation (22):$\begin{matrix}{{{Uadp}(k)} = {\frac{- 1}{{s1} \cdot {b1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}\left( {{{\overset{\_}{\sigma}(i)} \cdot \Delta}\quad T} \right)}}} & (27)\end{matrix}$

The latest identified gain coefficients a1(k) hat, a2(k) hat, b1(k) hatwhich have been determined by the identifier 25 are basically used asthe gain coefficients a1, a1, b1 that are required to calculate theequivalent control input Ueq, the reaching law input Urch, and theadaptive law input Uadp according to the equations (24), (26), (27). Thevalues of the switching function a bar in each control cycle which arerequired to calculate the reaching law input Urch and the adaptive lawinput Uadp are represented by the latest estimated differential outputVO2(k+1) bar determined by the estimator 26 and the estimateddifferential output VO2(k+d−1) bar determined by the estimator 26 in thepreceding control cycle.

The sliding mode controller 27 determines the sum of the equivalentcontrol input Ueq, the reaching law input Urch, and the adaptive lawinput Uadp determined according to the equations (24), (26), (27), asthe SLD manipulating input Usl to be applied to the exhaust system E(see the equation (18)). The conditions for establishing thecoefficients s1, s2, F, G used in the equations (24), (26), (27) are asdescribed above.

The above process is a basic algorithm for the sliding mode controller27 to determine the SLD manipulating input Usl (=target differentialair-fuel ratio kcmd) to be applied to the exhaust system E. According tothe above algorithm, the SLD manipulating input Usl is determined toconverge the estimated differential output VO2 bar from the O₂ sensor 6to “0” (as a result, to converge the output VO2/OUT from the O₂ sensor 6to the target value VO2/TARGET).

The sliding mode controller 27 eventually sequentially determines thetarget air-fuel ratio KCMD in each control cycle. The SLD manipulatinginput Usl determined as described above signifies a target value for thedifference between the upstream-of-catalyst air-fuel ratio detected bythe LAF sensor 5 and the air-fuel ratio reference value FLAF/BASE, i.e.,the target differential air-fuel ratio kcmd. Consequently, the slidingmode controller 27 eventually determines the target air-fuel ratio KCMDby adding the reference value FLAF/BASE to the determined SLDmanipulating input Usl in each control cycle according to the followingequation (28): $\quad\begin{matrix}\begin{matrix}{{{KCMD}(k)} = {{{Us1}(k)} + {{FLAF}/{BASE}}}} \\{= {{{Ueq}(k)} + {{Urch}(k)} + {{Uadp}(k)} + {{FLAF}/{BASE}}}}\end{matrix} & (28)\end{matrix}$

The above process is a basic algorithm for the sliding mode controller27 to sequentially determine the target air-fuel ratio KCMD according tothe present embodiment.

In the present embodiment, the stability of the adaptive sliding modecontrol process carried out by the sliding mode controller 27 is checkedfor limiting the value of the SLD manipulating input Usl. Details ofsuch a checking process will be described later on.

The general feedback controller 15 of the engine-side control unit 7 b,particularly, the adaptive controller 18, will further be describedbelow.

In FIG. 1, the general feedback controller 15 effects a feedback controlprocess to converge the output KACT from the LAF sensor 5 to the targetair-fuel ratio KCMD as described above. If such a feedback controlprocess were carried out under the known PID control only, it would bedifficult to keep stable controllability against dynamic behavioralchanges including changes in the operating state of the internalcombustion engine 1, characteristic changes due to aging of the internalcombustion engine 1, etc.

The adaptive controller 18 is a recursive-type controller which makes itpossible to carry out a feedback control process while compensating fordynamic behavioral changes of the internal combustion engine 1. As shownin FIG. 7, the adaptive controller 18 comprises a parameter adjuster 30for establishing a plurality of adaptive parameters using the parameteradjusting law proposed by I. D. Landau, et al., and a manipulatedvariable calculator 31 for calculating the feedback manipulated variableKSTR using the established adaptive parameters.

The parameter adjuster 30 will be described below. According to theadjusting law proposed by I. D. Landau, et al., when polynomials of thedenominator and the numerator of a transfer function B(Z⁻¹)/A(Z⁻¹) of adiscrete-system object to be controlled are generally expressedrespectively by equations (29), (30), given below, an adaptive parameterθ hat (j) (j indicates the ordinal number of a control cycle of theengine-side control unit 7 b) established by the parameter adjuster 30is represented by a vector (transposed vector) according to the equation(31) given below. An input ζ(j) to the parameter adjuster 30 isexpressed by the equation (32) given below. In the present embodiment,it is assumed that the internal combustion engine 1, which is an objectto be controlled by the general feedback controller 15, is considered tobe a plant of a first-order system having a dead time dp (the time ofthree combustion cycles of the internal combustion engine 1), and m=n=1,dp=3 in the equations (29)-(32), and five adaptive parameters s0, r1,r2, r3, b0 are established (see FIG. 7). In the upper and middleexpressions of the equation (34), us, ys generally represent an input(manipulated variable) to the object to be controlled and an output(controlled variable) from the object to be controlled. In the presentembodiment, the input is the feedback manipulated variable KSTR and theoutput from the object to be controlled (the internal combustion engine1) is the output KACT (upstream-of-catalyst air-fuel ratio) from the LAFsensor 5, and the input ζ(j) to the parameter adjuster 30 is expressedby the lower expression of the equation (32) (see FIG. 7).A(Z ⁻¹)=1+a 1 Z ⁻¹ + . . . +anZ ^(−n)  (29)B(Z ⁻¹)=b 0+b 1 Z ⁻¹ + . . . +bmZ ^(−m)  (30)$\begin{matrix}{\quad\begin{matrix}{{{\hat{\theta}}^{T}(j)} = \left\lbrack {{\hat{b}0(j)},{\hat{B}{R\left( {Z^{- 1},j} \right)}},{\hat{S}\left( {Z^{- 1},j} \right)}} \right\rbrack} \\{= \left\lbrack {{{b0}(j)},{{r1}(j)},\ldots\quad,{{rm} + {dp} - {1(j)}},{{s0}(j)},\ldots\quad,{{sn} - {1(j)}}} \right\rbrack} \\{= \left\lbrack {{{b0}(j)},{{r1}(j)},{{r2}(j)},{{r3}(j)},{{s0}(j)}} \right\rbrack}\end{matrix}} & (31) \\\begin{matrix}{{\zeta^{T}(j)} = \left\lbrack {{{us}(j)},\ldots\quad,{{us}\left( {j - m - {dp} + 1} \right)},{{ys}(j)},\ldots\quad,{{ys}\left( {j - n + 1} \right)}} \right\rbrack} \\{= \left\lbrack {{{us}(j)},{{us}\left( {j - 1} \right)},{{us}\left( {j - 2} \right)},{{us}\left( {j - 3} \right)},{{ys}(j)}} \right\rbrack} \\{= \left\lbrack {{{KSTR}(j)},{{KSTR}\left( {j - 1} \right)},{{KSTR}\left( {j - 2} \right)},{{KSTR}\left( {j - 3} \right)},{{KACT}(j)}} \right\rbrack}\end{matrix} & (32)\end{matrix}$

The adaptive parameter θ hat expressed by the equation (31) is made upof a scalar quantity element b0 hat (j) for determining the gain of theadaptive controller 18, a control element BR hat (Z⁻¹,j) expressed usinga manipulated variable, and a control element S (Z⁻¹,j) expressed usinga controlled variable, which are expressed respectively by the followingequations (33) through (35) (see the block of the manipulated variablecalculator 31 shown in FIG. 7); $\begin{matrix}{{\hat{b}0^{- 1}(j)} = \frac{1}{b0}} & (33) \\\begin{matrix}{{\hat{B}{R\left( {Z^{- 1},j} \right)}} = {{r1Z}^{- 1} + {r2Z}^{- 2} + \ldots + {rm} + {dp} - {1Z^{- {({n + {dp} - 1})}}}}} \\{= {{r1Z}^{- 1} + {r2Z}^{- 2} + {r3Z}^{- 3}}}\end{matrix} & (34) \\\begin{matrix}{{\hat{S}\left( {Z^{- 1},j} \right)} = {{s0} + {s1Z}^{- 1} + \ldots + {sn} - {1Z^{- {({n - 1})}}}}} \\{= {s0}}\end{matrix} & (35)\end{matrix}$

The parameter adjuster 30 establishes coefficients of the scalarquantity element and the control elements, described above, and suppliesthem as the adaptive parameter θ hat expressed by the equation (31) tothe manipulated variable calculator 31. The parameter adjuster 30calculates the adaptive parameter θ hat so that the output KACT from theLAF sensor 5 will agree with the target air-fuel ratio KCMD, usingtime-series data of the feedback manipulated variable KSTR from thepresent to the past and the output KACT from the LAF sensor 5.

Specifically, the parameter adjuster 30 calculates the adaptiveparameter θ hat according to the following equation (36):{circumflex over (θ)}(j)={circumflex over(θ)}(j−1)+Γ(j−1)·ζ(j−dp)·e*(j)  (36)where Γ(j) represents a gain matrix (whose degree is indicated bym+n+dp) for determining a rate of establishing the adaptive parameter θhat, and e*(j) an estimated error of the adaptive parameter θ hat. Γ(j)and e*(j) are expressed respectively by the following recursive formulas(37), (38): $\begin{matrix}{{{\Gamma(j)} = {{{\frac{1}{{\lambda 1}(j)}\left\lbrack {{\Gamma\left( {j - 1} \right)} - \frac{{{\lambda 2}(j)} \cdot {\Gamma\left( {j - 1} \right)} \cdot {\zeta\left( {j - {dp}} \right)} \cdot {\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma\left( {j - 1} \right)}}{{{\lambda 1}(j)} + {{{\lambda 2}(j)} \cdot {\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma\left( {j - 1} \right)} \cdot {\zeta\left( {j - {dp}} \right)}}}} \right\rbrack}\quad{where}\quad 0} < {{\lambda 1}(j)} \leq 1}},{0 \leq {{\lambda 2}(j)} < 2},{{\Gamma(0)} > 0.}} & (37) \\{{e*(j)} = \frac{{{D\left( Z^{- 1} \right)} \cdot {{KACT}(j)}} - {{{\hat{\theta}}^{T}\left( {j - 1} \right)} \cdot {\zeta\left( {j - {dp}} \right)}}}{1 + {{\zeta^{T}\left( {j - {dp}} \right)} \cdot {\Gamma\left( {j - 1} \right)} \cdot {\zeta\left( {j - {dp}} \right)}}}} & (38)\end{matrix}$where D(Z⁻¹) represents an asymptotically stable polynomial foradjusting the convergence. In the present embodiment, D(Z⁻¹)=1.

Various specific algorithms including the degressive gain algorithm, thevariable gain algorithm, the fixed tracing algorithm, and the fixed gainalgorithm are obtained depending on how λ1(j), λ2(j) in the equation(37) are selected. For a time-dependent plant such as a fuel injectionprocess, an air-fuel ratio, or the like of the internal combustionengine 1, either one of the degressive gain algorithm, the variable gainalgorithm, the fixed gain algorithm, and the fixed tracing algorithm issuitable.

Using the adaptive parameter θ hat (s0, r1, r2, r3, b0) established bythe parameter adjuster 30 and the target air-fuel ratio KCMD determinedby the target air-fuel ratio generation processor 13, the manipulatedvariable calculator 31 determines the feedback manipulated variable KSTRaccording to a recursive formula expressed by the following equation(39): $\begin{matrix}{{KSTR} = \frac{\begin{matrix}{{{KCMD}(j)} - {{S0} \cdot {{KACT}(j)}} - {{{r1} \cdot {KSTR}}\left( {j - 1} \right)} -} \\{{{{r2} \cdot {KSTR}}\left( {j - 2} \right)} - {{r3} \cdot {{KSTR}\left( {j - 3} \right)}}}\end{matrix}}{b0}} & (39)\end{matrix}$The manipulated variable calculator 31 shown in FIG. 7 represents ablock diagram of the calculations according to the equation (39).

The feedback manipulated variable KSTR determined according to theequation (39) becomes the target air-fuel ratio KCMD insofar as theoutput KACT of the LAF sensor 5 agrees with the target air-fuel ratioKCMD. Therefore, the feedback manipulated variable KSTR is divided bythe target air-fuel ratio KCMD by the divider 19 for thereby determiningthe feedback manipulated variable kstr that can be used as the feedbackcorrection coefficient KFB.

As is apparent from the foregoing description, the adaptive controller18 thus constructed is a recursive-type controller taking into accountdynamic behavioral changes of the internal combustion engine 1 which isan object to be controlled. Stated otherwise, the adaptive controller 18is a controller described in a recursive form to compensate for dynamicbehavioral changes of the internal combustion engine 1, and moreparticularly a controller having a recursive-type adaptive parameteradjusting mechanism.

A recursive-type controller of this type may be constructed using anoptimum regulator. In such a case, however, it generally has noparameter adjusting mechanism. The adaptive controller 18 constructed asdescribed above is suitable for compensating for dynamic behavioralchanges of the internal combustion engine 1.

The details of the adaptive controller 18 have been described above.

The PID controller 17, which is provided together with the adaptivecontroller 18 in the general feedback controller 15, calculates aproportional term (P term), an integral term (I term), and a derivativeterm (D term) from the difference between the output KACT of the LAFsensor 5 and the target air-fuel ratio KCMD, and calculates the total ofthose terms as the feedback manipulated variable KLAF, as is the casewith the general PID control process. In the present embodiment, thefeedback manipulated variable KLAF is set to “1” when the output KACT ofthe LAF sensor 5 agrees with the target air-fuel ratio KCMD by settingan initial value of the integral term (I term) to “1”, so that thefeedback manipulated variable KLAF can be used as the feedbackcorrection coefficient KFB for directly correcting the fuel injectionquantity. The gains of the proportional term, the integral term, and thederivative term are determined from the rotational speed and intakepressure of the internal combustion engine 1 using a predetermined map.

The switcher 20 of the general feedback controller 15 outputs thefeedback manipulated variable KLAF determined by the PID controller 17as the feedback correction coefficient KFB for correcting the fuelinjection quantity if the combustion in the internal combustion engine 1tends to be unstable as when the temperature of the coolant of theinternal combustion engine 1 is low, the internal combustion engine 1rotates at high speeds, or the intake pressure is low, or if the outputKACT of the LAF sensor 5 is not reliable due to a response delay of theLAF sensor 5 as when the target air-fuel ratio KCMD changes largely orimmediately after the air-fuel ratio feedback control process hasstarted, or if the internal combustion engine 1 operates highly stablyas when it is idling and hence no high-gain control process by theadaptive controller 18 is required. Otherwise, the switcher 20 outputsthe feedback manipulated variable kstr which is produced by dividing thefeedback manipulated variable KSTR determined by the adaptive controller18 by the target air-fuel ration KCMD, as the feedback correctioncoefficient KFB for correcting the fuel injection quantity. This isbecause the adaptive controller 18 effects a high-gain control processand functions to converge the output KACT of the LAF sensor 5 quickly tothe target air-fuel ratio KCMD, and if the feedback manipulated variableKSTR determined by the adaptive controller 18 is used when thecombustion in the internal combustion engine 1 is unstable or the outputKACT of the LAF sensor 5 is not reliable, then the air-fuel ratiocontrol process tends to be unstable.

Such operation of the switcher 20 is disclosed in detail in Japaneselaid-open patent publication No. 8-105345 or U.S. Pat. No. 5,558,075 bythe applicant of the present application, and will not be described indetail below.

Operation of the apparatus according to the present embodiment will bedescribed below.

First, a process carried out by the engine-side control unit 7 b will bedescribed below with reference to FIG. 8. The engine-side control unit 7b calculates an output fuel injection quantity #nTout for each of thecylinders in synchronism with a crankshaft angle period (TDC) of theinternal combustion engine 1 as follows:

The engine-side control unit 7 b reads outputs from various sensorsincluding the LAF sensor 5 and the O₂ sensor 6 in STEPa. At this time,the output KACT of the LAF sensor 5 and the output VO2/OUT of the O₂sensor 6, including data obtained in the past, are stored in atime-series fashion in a memory (not shown).

Then, the basic fuel injection quantity calculator 8 corrects a fuelinjection quantity corresponding to the rotational speed NE and intakepressure PB of the internal combustion engine 1 depending on theeffective opening area of the throttle valve, thereby calculating abasic fuel injection quantity Tim in STEPb. The first correctioncoefficient calculator 9 calculates a first correction coefficientKTOTAL depending on the coolant temperature and the amount by which thecanister is purged in STEPc.

The engine-side control unit 7 b decides whether the operation mode ofthe internal combustion engine 1 is an operation mode (hereinafterreferred to as “normal operation mode”) in which the fuel injectionquantity is adjusted using the target air-fuel ratio KCMD generated bythe target air-fuel ratio generation processor 13, and sets a value of aflag f/prism/on in STEPd. When the value of the flag f/prism/on is “1”,it means that the operation mode of the internal combustion engine 1 isthe normal operation mode, and when the value of the flag f/prism/on is“0”, it means that the operation mode of the internal combustion engine1 is not the normal operation mode.

The deciding subroutine of STEPd is shown in detail in FIG. 9. As shownin FIG. 9, the engine-side control unit 7 b decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEPd-1, STEPd-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the target air-fuel ratio generation processor 13are not accurate enough, the operation mode of the internal combustionengine 1 is not the normal operation mode, and the value of the flagf/prism/on is set to “0” in STEPd-10.

Then, the engine-side control unit 7 b decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEPd-3. The engine-side control unit 7 b decides whether the ignitiontiming of the internal combustion engine 1 is retarded for earlyactivation of the catalytic converter 3 immediately after the start ofthe internal combustion engine 1 or not in STEPd-4. The engine-sidecontrol unit 7 b decides whether the throttle valve of the internalcombustion engine 1 is substantially fully open or not in STEPd-5. Theengine-side control unit 7 b decides whether the supply of fuel to theinternal combustion engine 1 is being stopped or not in STEPd-6. Ifeither one of the conditions of these steps is satisfied, then since itis not preferable or not possible to control the supply of fuel to theinternal combustion engine 1 using the target air-fuel ratio KCMDgenerated by the target air-fuel ratio generation processor 13, theoperation mode of the internal combustion engine 1 is not the normaloperation mode, and the value of the flag f/prism/on is set to “0” inSTEPd-10.

The engine-side control unit 7 b then decides whether the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1fall within respective given ranges (normal ranges) or not respectivelyin STEPd-7, STEPd-8. If either one of the rotational speed NE and theintake pressure PB does not fall within its given range, then since itis not preferable to control the supply of fuel to the internalcombustion engine 1 using the target air-fuel ratio KCMD generated bythe target air-fuel ratio generation processor 13, the operation mode ofthe internal combustion engine 1 is not the normal operation mode, andthe value of the flag f/prism/on is set to “0” in STEPd-10.

If the conditions of STEPd-1, STEPd-2, STEPd-7, STEPd-8 are satisfied,and the conditions of STEPd-3, STEPd-4, STEPd-5, STEPd-6 are notsatisfied (at this time, the internal combustion engine 1 is in thenormal operation mode), then the operation mode of the internalcombustion engine 1 is judged as the normal operation mode, and thevalue of the flag f/prism/on is set to “1” in STEPd-9.

In FIG. 8, after the value of the flag f/prism/on has been set, theengine-side control unit 7 b determines the value of the flag f/prism/onin STEPe. If f/prism/on=1, then the engine-side control unit 7 b readsthe target air-fuel ratio KCMD generated by the exhaust-side mainprocessor 13 in STEPf. If f/prism/on=0, then the engine-side controlunit 7 b sets the target air-fuel ratio KCMD to a predetermined value inSTEPg. The predetermined value to be established as the target air-fuelratio KCMD is determined from the rotational speed NE and intakepressure PB of the internal combustion engine 1 using a predeterminedmap, for example.

In the local feedback controller 16, the PID controller 22 calculatesrespective feedback correction coefficients #nKLAF in order to eliminatevariations between the cylinders, based on actual air-fuel ratios #nA/Fof the respective cylinders which have been estimated from the outputKACT of the LAF sensor 5 by the observer 21, in STEPh. Then, the generalfeedback controller 15 calculates a feedback correction coefficient KFBin STEPi.

Depending on the operating state of the internal combustion engine 1,the switcher 20 selects either the feedback manipulated variable KLAFdetermined by the PID controller 17 or the feedback manipulated variablekstr which has been produced by dividing the feedback manipulatedvariable KSTR determined by the adaptive controller 18 by the targetair-fuel ratio KCMD (normally, the switcher 20 selects the feedbackmanipulated variable kstr from the adaptive controller 18). The switcher20 then outputs the selected feedback manipulated variable KLAF or kstras a feedback correction coefficient KFB for correcting the fuelinjection quantity.

When switching the feedback correction coefficient KFB from the feedbackmanipulated variable KLAF from the PID controller 17 to the feedbackmanipulated variable kstr from the adaptive controller 18, the adaptivecontroller 18 determines a feedback manipulated variable KSTR in amanner to hold the correction coefficient KFB to the precedingcorrection coefficient KFB (=KLAF) as long as in the cycle time for theswitching. When switching the feedback correction coefficient KFB fromthe feedback manipulated variable kstr from the adaptive controller 18to the feedback manipulated variable KLAF from the PID controller 17,the PID controller 17 calculates a present correction coefficient KLAFin a manner to regard the feedback manipulated variable KLAF determinedby itself in the preceding cycle time as the preceding correctioncoefficient KFB (=kstr).

After the feedback correction coefficient KFB has been calculated, thesecond correction coefficient calculator 10 calculates in STEPj a secondcorrection coefficient KCMDM depending on the target air-fuel ratio KCMDdetermined in STEPf or STEPg.

Then, the engine-side control unit 7 b multiplies the basic fuelinjection quantity Tim determined as described above, by the firstcorrection coefficient KTOTAL, the second correction coefficient KCMDM,the feedback correction coefficient KFB, and the feedback correctioncoefficients #nKLAF of the respective cylinders, determining output fuelinjection quantities #nTout of the respective cylinders in STEPk. Theoutput fuel injection quantities #nTout are then corrected foraccumulated fuel particles on intake pipe walls of the internalcombustion engine 1 by the fuel accumulation corrector 23 in STEPm. Thecorrected output fuel injection quantities #nTout are output to thenon-illustrated fuel injectors of the internal combustion engine 1 inSTEPn. In the internal combustion engine 1, the fuel injectors injectfuel into the respective cylinders according to the respective outputfuel injection quantities #nTout.

The above calculation of the output fuel injection quantities #nTout andthe fuel injection of the internal combustion engine 1 are carried outin successive cycle times synchronous with the crankshaft angle periodof the internal combustion engine 1 for controlling the air-fuel ratioof the internal combustion engine 1 in order to converge the output KACTof the LAF sensor 5 (the detected value of the upstream-of-catalystair-fuel ratio) to the target air-fuel ratio KCMD. While the feedbackmanipulated variable kstr from the adaptive controller 18 is being usedas the feedback correction coefficient KFB, the output KACT of the LAFsensor 5 is quickly converged to the target air-fuel ratio KCMD withhigh stability against behavioral changes such as changes in theoperating state of the internal combustion engine 1 or characteristicchanges thereof. A response delay of the internal combustion engine 1 isalso appropriately compensated for.

Concurrent with the above fuel control for the internal combustionengine 1, the exhaust-side control unit 7 a executes a flowchart of FIG.13 in control cycles of a constant period.

As shown in FIG. 13, the exhaust-side control unit 7 a decides whetherthe processing of the target air-fuel ratio generation processor 13(specifically, the processing of the identifier 25, the estimator 26,and the sliding mode controller 27) is to be executed or not, and sets avalue of a flag f/prim/cal indicative of whether the processing is to beexecuted or not in STEP1. When the value of the flag f/prim/cal is “0”,it means that the processing of the target air-fuel ratio generationprocessor 13 is not to be executed, and when the value of the flagf/prim/cal is “1”, it means that the processing of the target air-fuelratio generation processor 13 is to be executed.

The deciding subroutine in STEP1 is shown in detail in FIG. 11. As shownin FIG. 11, the exhaust-side control unit 7 a decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEP1-1, STEP1-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the target air-fuel ratio generation processor 13are not accurate enough, the value of the flag f/prism/cal is set to “0”in STEP1-6. Then, in order to initialize the identifier 25 as describedlater on, the value of a flag f/id/reset indicative of whether theidentifier 25 is to be initialized or not is set to “1” in STEP1-7. Whenthe value of the flag f/id/reset is “1”, it means that the identifier 25is to be initialized, and when the value of the flag f/id/reset is “0”,it means that the identifier 25 is not to be initialized.

The exhaust-side control unit 7 a decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEP1-3. The exhaust-side control unit 7 a decides whether the ignitiontiming of the internal combustion engine 1 is retarded for earlyactivation of the catalytic converter 3 immediately after the start ofthe internal combustion engine 1 or not in STEP1-4. If the conditions ofthese steps are satisfied, then since the target air-fuel ratio KCMDcalculated to adjust the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET is not used for the fuel control for the internalcombustion engine 1, the value of the flag f/id/cal is set to “0”inSTEP1-6, and the value of the flag f/id/reset is set to “1” in order toinitialize the identifier 25 in STEP1-7.

In FIG. 10, after the above deciding subroutine, the exhaust-sidecontrol unit 7 a decides whether a process of identifying (updating) thegain coefficients a1, a1, b1 with the identifier 25 is to be executed ornot, and sets a value of a flag f/id/cal indicative of whether theprocess of identifying (updating) the gain coefficients a1, a1, b1 is tobe executed or not in STEP2. When the value of the flag f/id/cal is “0”,it means that the process of identifying (updating) the gaincoefficients a1, a1, b1 is not to be executed, and when the value of theflag f/id/cal is “1”, it means that the process of identifying(updating) the gain coefficients a1, a1, b1 is to be executed.

In the deciding process of STEP2, the exhaust-side control unit 7 adecides whether the throttle valve of the internal combustion engine 1is substantially fully open or not, and also decides whether the supplyof fuel to the internal combustion engine 1 is being stopped or not. Ifeither one of these conditions is satisfied, then since it is difficultto identify the gain coefficients a1, a1, b1 appropriately, the value ofthe flag f/id/cal is set to “0”. If neither one of these conditions issatisfied, then the value of the flag f/id/cal is set to “1” to identify(update) the gain coefficients a1, a1, b1 with the identifier 25.

The flow rate data generating means 28 calculates an estimated exhaustgas volume ABSV according to the equation (3) from the latest detectedvalues (acquired by the engine-side control unit 7 b in STEPa in FIG. 8)of the present rotational speed NE and intake pressure PB of theinternal combustion engine 1 in STEP3. Thereafter, the dead time settingmeans 29 determines the values of respective set dead times d1, d2 ofthe exhaust system E and the air-fuel ratio manipulating system from thecalculated value of the estimated exhaust gas volume ABSV according tothe data table shown in FIG. 4 in STEP4. The values of the set deadtimes d1, d2 determined in STEP4 are integral values which are producedby rounding off the fractions of the values determined from the datatable shown in FIG. 4, as described above.

Then, the exhaust-side control unit 7 a calculates the latestdifferential outputs kact(k) (=KACT−FLAF/BASE), VO2(k)(=VO2/OUT−VO2/TARGET) respectively with the subtractors 11, 12 in STEP5.Specifically, the subtractors 11, 12 select latest ones of thetime-series data read and stored in the non-illustrated memory in STEPashown in FIG. 8, and calculate the differential outputs kact(k), VO2(k).The differential outputs kact(k), VO2(k), as well as data given in thepast, are stored in a time-series manner in the non-illustrated memoryin the exhaust-side control unit 7 a.

Then, in STEP6, the exhaust-side control unit 7 a determines the valueof the flag f/prism/cal set in STEP1. If the value of the flagf/prism/cal is “0”, i.e., if the processing of the target air-fuel ratiogeneration processor 13 is not to be executed, then the exhaust-sidecontrol unit 7 a forcibly sets the SLD manipulating input Usl (thetarget differential air-fuel ratio kcmd) to be determined by the slidingmode controller 27, to a predetermined value in STEP14. Thepredetermined value may be a fixed value (e.g., “0”) or the value of theSLD manipulating input Usl determined in a preceding control cycle.

After the SLD manipulating input Usl is set to the predetermined value,the exhaust-side control unit 7 a adds the reference value FLAF/BASE tothe SLD manipulating input Usl for thereby determining a target air-fuelratio KCMD in the present control cycle in STEP 15. Then, the processingin the present control cycle is finished.

If the value of the flag f/prism/cal is “1” in STEP6, i.e., if theprocessing of the target air-fuel ratio generation processor 13 is to beexecuted, then the exhaust-side control unit 7 a effects the processingof the identifier 25 in STEP7.

The processing of the identifier 25 is carried out according to aflowchart shown in FIG. 12. The identifier 25 determines the value ofthe flag f/id/cal set in STEP2 in STEP7-1. If the value of the flagf/id/cal is “0”, then since the process of identifying the gaincoefficients a1, a1, b1 with the identifier 25 is not carried out,control immediately goes back to the main routine shown in FIG. 10.

If the value of the flag f/id/cal is “1”, then the identifier 25determines the value of the flag f/id/reset set in STEP1 with respect tothe initialization of the identifier 25 in STEP7-2. If the value of theflag f/id/reset is “1”, the identifier 25 is initialized in STEP7-3.When the identifier 25 is initialized, the identified gain coefficientsa1 hat, a2 hat, b1 hat are set to predetermined initial values (theidentified gain coefficient vector Θ according to the equation (5) isinitialized), and the elements of the matrix P (diagonal matrix)according to the equation (11) are set to predetermined initial values.The value of the flag f/id/reset is reset to “0”.

Then, the identifier 25 determines the value of the weighted parameterλ₁ in the algorithm of the method of weighted least squares of theidentifier 25, i.e., the value of the weighted parameter λ₁ used in theequation (11), from the present value of the estimated exhaust gasvolume ABSV determined by the flow rate data generating means 28 inSTEP3 according to the data table shown in FIG. 5 in STEP7-4.

Then, the identifier 25 calculates the identified differential outputVO2(k) hat using the values of the present identified gain coefficientsa1(k−1) hat, a2(k−1) hat, b1(k−1) hat and the past data VO2(k−1),VO2(k−2), kact(k-d1−1) of the differential outputs VO2, kact calculatedin each control cycle in STEP5, according to the equation (4) inSTEP7-5. Specifically, the differential output kact(k-d1−1) used in theabove calculation is a differential output kact at a past timedetermined by the set dead time d1 of the exhaust system E that is setby the dead time setting means 29 in STEP4, and also a differentialoutput kact obtained in a control cycle that is (d1+1) control cyclesprior to the present control cycle.

The identifier 25 then calculates the vector Kθ(k) to be used indetermining the new identified gain coefficients a1 hat, a2 hat, b1 hataccording to the equation (10) in STEP7-6. Thereafter, the identifier 25calculates the identified error id/e(k) (the difference between theidentified differential output VO2 hat and the actual differentialoutput VO2, see the equation (8)), in STEP7-7.

The identified error id/e(k) may basically be calculated according tothe equation (8). In the present embodiment, however, a value(=VO2(k)-VO2(k) hat) calculated according to the equation (8) from thedifferential output VO2 calculated in each control cycle in STEP3, andthe identified differential output VO2 hat calculated in each controlcycle in STEP7-5 is filtered with low-pass characteristics to calculatethe identified error id/e(k).

This is because since the behavior of the exhaust system E including thecatalytic converter 3 generally has low-pass characteristics, it ispreferable to attach importance to the low-frequency behavior of theexhaust system E in appropriately identifying the gain coefficients a1,a2, b1 of the exhaust system model.

Both the differential output VO2 and the identified differential outputVO2 hat may be filtered with the same low-pass characteristics. Forexample, after the differential output VO2 and the identifieddifferential output VO2 hat have separately been filtered, the equation(7) may be calculated to determine the identified error id/e(k). Theabove filtering is carried out by a moving average process which is adigital filtering process.

Thereafter, the identifier 25 calculates a new identified gaincoefficient vector Θ(k), i.e., new identified gain coefficients a1(k)hat, a2(k) hat, b1(k) hat, according to the equation (9) using theidentified error id/e(k) determined in STEP7-7 and Kθ calculated inSETP7-6 in STEP7-8.

After having calculated the new identified gain coefficients a1(k) hat,a2(k) hat, b1(k) hat, the identifier 25 limits the values of the gaincoefficients a1 hat, a2 hat, b1 hat within a predetermined range asdescribed below in STEP7-9. Then, the identifier 25 updates the matrixP(k) according to the equation (11) for the processing of a next controlcycle in STEP7-10, after which control returns to the main routine shownin FIG. 10.

The process of limiting the identified gain coefficients a1 hat, a2 hat,b1 hat in STEP7-9 comprises a process of eliminating the situation wherethe target air-fuel ratio KCMD determined by the sliding mode controller27 varies in a high-frequency oscillating manner. The inventors of thepresent invention have found that if the values of the identified gaincoefficients a1 hat, a2 hat, b1 hat are not particularly limited, whilethe output signal VO2/OUT of the O₂ sensor 6 is being stably controlledat the target value VO2/TARGET, there are developed a situation in whichthe target air-fuel ratio KCMD determined by the sliding mode controller27 changes smoothly with time, and a situation in which the targetair-fuel ratio KCMD oscillates with time at a high frequency. Whetherthe target air-fuel ratio KCMD changes smoothly or oscillates at a highfrequency depends on the combinations of the values of the identifiedgain coefficients a1 hat, a2 hat relative to the response delay elementof the exhaust system model (more specifically, the primaryautoregressive term and the secondary autoregressive term on the rightside of the equation (1)) and the value of the identified gaincoefficient b1 hat relative to the dead time element of the exhaustsystem model.

The limiting process in STEP7-9 is roughly classified into a process oflimiting the combination of the values of the identified gaincoefficients a1 hat, a2 hat within a given range, and a process oflimiting the value of the identified gain coefficient b1 hat within agiven range.

The range within which the combination of the values of the identifiedgain coefficients a1 hat, a2 hat are limited and the range within whichthe value of the identified gain coefficient b1 hat is limited areestablished as follows:

With respect to the range within which the combination of the values ofthe identified gain coefficients a1 hat, a2 hat are limited, a studymade by the inventors indicates that whether the target air-fuel ratioKCMD changes smoothly or oscillates at a high frequency is closelyrelated to combinations of the coefficient values α1, α2 used for theestimator 26 to determine the estimated differential output VO2(k+d) bar(these coefficient values α1, α2 are the first-row, first-column elementand the first-row, second-column element of the matrix A^(d) which is apower of the matrix A defined by the equation (13)).

Specifically, as shown in FIG. 13, when a coordinate plane whosecoordinate components are represented by the coefficient values α1, α2is established, if a point on the coordinate plane which is determinedby a combination of the coefficient values α1, α2 lies in a hatchedrange, which is surrounded by a triangle Q₁Q₂Q₃ (including theboundaries) and will hereinafter be referred to as an estimatingcoefficient stable range, then the target air-fuel ratio KCMD tends tobe smooth. Conversely, if a point determined by a combination of thecoefficient values α1, α2 lies outside of the estimating coefficientstable range, then the target air-fuel ratio KCMD is liable to oscillatewith time at a high frequency or the controllability of the outputVO2/OUT of the O₂ sensor 6 at the target value VO2/TARGET is liable tobecome poor.

Therefore, the combinations of the values of the gain coefficients a1,a2 should be limited such that the point on the coordinate plane shownin FIG. 13 which corresponds to the combination of the coefficientvalues α1, α2 determined by the values of the identified gaincoefficients a1 hat, a2 hat will lie within the estimating coefficientstable range.

In FIG. 13, a triangular range Q₁Q₄Q₃ on the coordinate plane whichcontains the estimating coefficient stable range is a range thatdetermines combinations of the coefficient values α1, α2 which makestheoretically stable a system defined according to the followingequation (40), i.e., a system defined by an equation similar to theequation (13) except that VO2(k), VO2(k−1) on the right side of theequation (13) are replaced respectively with VO2(k) bar, VO2(k−1) bar(VO2(k) bar, VO2(k−1) bar mean respectively an estimated differentialoutput determined in each control cycle by the estimator 26 and anestimated differential output determined in a preceding cycle by theestimator 26). $\begin{matrix}{{\overset{\_}{VO2}\left( {k + d} \right)} = {{{\alpha 1} \cdot {\overset{\_}{VO2}(k)}} + {{\alpha 2} \cdot {\overset{\_}{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{d}{\beta_{j} \cdot {{kcmd}\left( {k - j} \right)}}}}} & (40)\end{matrix}$

The condition for the system defined according to the equation (40) tobe stable is that a pole of the system (which is given by the followingequation (41)) exists in a unit circle on a complex plane:

Pole of the system according to the equation $\begin{matrix}{= \frac{{\alpha 1} \pm \sqrt{{\alpha 1}^{2} + {4 \cdot {\alpha 2}}}}{2}} & (41)\end{matrix}$

The triangular range Q₁Q₄Q₃ shown in FIG. 13 is a range for determiningthe combinations of the coefficient values α1, α2 which satisfy theabove condition. Therefore, the estimating coefficient stable range is arange indicative of those combinations where α1≧0 of the combinations ofthe coefficient values α1, α2 which make stable the system defined bythe equation (40).

Since the coefficient values α1, α2 are determined by a combination ofthe values of the gain coefficients α1, α2 when the total set dead timed is determined to be of a certain value, a combination of the values ofthe gain coefficients α1, α2 is determined from a combination of thecoefficient values α1, α2 using the value of the total set dead time d.Therefore, the estimating coefficient stable range shown in FIG. 13which determines preferable combinations of the coefficient values α1,α2 can be converted into a range on a coordinate plane shown in FIG. 14whose coordinate components are represented by the gain coefficients al,a2.

If the above conversion is carried out with the total set dead time dbeing determined to be of a certain value, then the estimatingcoefficient stable range is converted into a range enclosed by theimaginary lines in FIG. 14, which is a substantially triangular rangehaving an undulating lower side and will hereinafter be referred to asan identifying coefficient stable range, on the coordinate plane shownin FIG. 14. Stated otherwise, when a point on the coordinate plane shownin FIG. 14 which is determined by a combination of the values of thegain coefficients a1, a2 resides in the identifying coefficient stablerange enclosed by the imaginary lines in FIG. 14, a point on thecoordinate plane shown in FIG. 13 which corresponds to the combinationof the coefficient values α1, α2 determined by those values of the gaincoefficients a1, a2 resides in the estimating coefficient stable range.The identifying coefficient stable range changes with the value of thetotal set dead time d, as described later on. It is assumed for a whilein the description below that the total set dead time d is fixed to acertain value (represented by dx).

Consequently, the combinations of the values of the identified gaincoefficients a1 hat, a2 hat determined by the identifier 25 shouldpreferably be limited within such a range that a point on the coordinateplane shown in FIG. 14 which is determined by those values of theidentified gain coefficients a1 hat, a2 hat reside in the identifyingcoefficient stable range.

However, since a boundary (lower side) of the identifying coefficientstable range indicated by the imaginary lines in FIG. 14 is of a complexundulating shape, a practical process for limiting the point on thecoordinate plane shown in FIG. 14 which is determined by the values ofthe identified gain coefficients a1 hat, a2 hat within the identifyingcoefficient stable range is liable to be complex.

For this reason, according to the present embodiment, the identifyingcoefficient stable range (the identifying coefficient stable rangecorresponding to the total set dead time dx) is substantiallyapproximated by a quadrangular range Q₅Q₆Q₇Q₈ enclosed by the solidlines in FIG. 14, which has straight boundaries and will hereinafter bereferred to as an identifying coefficient limiting range. As shown inFIG. 14, the identifying coefficient limiting range (the identifyingcoefficient limiting range corresponding to the total set dead time dx)is a range enclosed by a polygonal line (including line segments Q₅Q₆and Q₅Q₈) expressed by a functional expression |a1|+a2=1, a straightline (including a line segment Q₆Q₇) expressed by a constant-valuedfunctional expression a1=A1L, and a straight line (including a linesegment Q₇Q₈) expressed by a constant-valued functional expressiona2=A2L. In the present embodiment, the identifying coefficient limitingrange is used as the range within which the combinations of the valuesof the identified gain coefficients a1 hat, a2 hat are limited. Althoughpart of the lower side of the identifying coefficient limiting rangedeviates from the identifying coefficient stable range, it hasexperimentally been confirmed that the point determined by theidentified gain coefficients a1 hat, a2 hat determined by the identifier25 does not actually fall in the deviating range. Therefore, thedeviating range will not pose any practical problem.

The identifying coefficient stable range which serves as a basis for theidentifying coefficient limiting range changes with the value of thetotal set dead time d, as is apparent from the definition of thecoefficient values α1, α2 according to the equation (13). In the presentembodiment, the values of the set dead time d1 of the exhaust system Eand the set dead time d2 of the air-fuel ratio manipulating system, andhence the value of the total set dead time d, are sequentially variablyset depending on the estimated exhaust gas volume ABSV.

The inventors have found that the identifying coefficient stable range,chiefly the shape of only its lower portion (generally an undulatingportion from Q7 to Q8 in FIG. 14), varies depending on the value of thetotal set dead time d, and as the value of the total set dead time d isgreater, the lower portion of the identifying coefficient stable rangetends to expand more downwardly (in the negative direction along the a2axis). The shape of the upper portion (generally a portion enclosed by atriangle Q5Q6Q8 in FIG. 14) of the identifying coefficient stable rangeis almost not affected by the value of the total set dead time d.

In the present embodiment, the lower limit value A2L of the gaincoefficient a1 in the identifying coefficient limiting range forlimiting the combinations of the values of the identified gaincoefficients a1 hat, a2 hat is variably set depending on the estimatedexhaust gas volume ABSV which determines the dead times d1, d2 of theexhaust system E and the air-fuel ratio manipulating system. The lowerlimit value A2L of the gain coefficient a1 is determined from the value(latest value) of the estimated exhaust gas volume ABSV based on apredetermined data table represented by the solid-line curve e in FIG.15, for example. The data table is determined such that as the value ofthe estimated exhaust gas volume ABSV is larger (as the total set deadtime d is shorter), the lower limit value A2L (<0) is smaller (theabsolute value is greater). Thus, the identifying coefficient limitingrange is established such that as the estimated exhaust gas volume ABSVis larger (as the total set dead time d is shorter), the identifyingcoefficient limiting range is expanded more downwardly. For example, ifthe value of the total set dead time d is longer than the value dxcorresponding to the identifying coefficient limiting range indicated bythe solid line in FIG. 14, then the lower portion of the identifyingcoefficient limiting range is expanded below the identifying coefficientlimiting range where d=dx.

The above identifying coefficient limiting range is given forillustrative purpose only, and may be equal to or may substantiallyapproximate the identifying coefficient stable range corresponding toeach value of the total set dead time d, or may be of any shape insofaras most or all of the identifying coefficient limiting range belongs tothe identifying coefficient stable range. Thus, the identifyingcoefficient limiting range may be established in various configurationsin view of the ease with which to limit the values of the identifiedgain coefficients a1 hat, a2 hat and the practical controllability. Forexample, while the boundary of an upper portion of the identifyingcoefficient limiting range is defined by the functional expression|a1|+a2=1 in the illustrated embodiment, combinations of the values ofthe gain coefficients a1, a2 which satisfy this functional expressionare combinations of theoretical stable limits where a pole of the systemdefined by the equation (40) exists on a unit circle on a complex plane.Therefore, the boundary of the upper portion of the identifyingcoefficient limiting range may be determined by a functional expression|a1|+a2=r (r is a value slightly smaller than “1” corresponding to thestable limits, e.g., 0.99) for higher control stability.

The range within which the value of the identified gain coefficient b1hat is limited is established as follows:

The inventors have found that the situation in which the time-dependingchange of the target air-fuel ratio KCMD is oscillatory at a highfrequency tends to happen also when the value of the identified gaincoefficient b1 hat is excessively large or small. Furthermore, the valueof the identified gain coefficient b1 hat which is suitable to cause thetarget air-fuel ratio KCMD to change smoothly with time is affected bythe total set dead time d, and tends to be greater as the total set deadtime d is shorter. According to the present embodiment, an upper limitvalue B1H and a lower limit value B1L (B1H>B1L>0) for determining therange of the gain coefficient b1 are sequentially established dependingon the value (latest value) of the estimated exhaust gas volume ABSVwhich determines the value of the total set dead time d, and the valueof the identified gain coefficient b1 hat is limited in a range that isdetermined by the upper limit value B1H and the lower limit value B1L.In the present embodiment, the upper limit value B1H and the lower limitvalue B1L which determine the range of the value of the gain coefficientb1 are determined based on data tables that are determined in advancethrough experimentation or simulation as indicated by the solid-linecurves f, g in FIG. 15. The data tables are basically established thatas the estimated exhaust gas volume ABSV is greater (as the total setdead time d is shorter), the upper limit value B1H and the lower limitvalue B1L are greater.

A process of limiting combinations of the values of the identified gaincoefficients a1 hat, a2 hat and the range of the value of the identifiedgain coefficient b1 is carried out as follows:

Referring to a flowchart shown in FIG. 16, the identifier 25 sets thelower limit value A2L of the gain coefficient a2 in the identifyingcoefficient limiting range and the upper limit value B1H and the lowerlimit value B1L of the gain coefficient b1 based on the data tablesshown in FIG. 15 from the latest value of the estimated exhaust gasvolume ABSV determined by the flow rate data generating means 28 inSTEP3 shown in FIG. 10, in STEP7-9-1.

The identifier 25 first limits combinations of the identified gaincoefficients a1(k) hat, a2(k) hat, of the identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat that have been determined in STEP7-8shown in FIG. 12, within the identifying coefficient limiting range inSTEP7-9-2 through STEP7-9-9.

Specifically, the identifier 25 decides whether or not the value of theidentified gain coefficient a2(k) hat determined in STEP7-8 is equal toor greater than the lower limit value A2L (see FIG. 14) set inSTEP7-9-1, in STEP7-9-2.

If A2(k) hat<A2L, then since a point on the coordinate plane shown inFIG. 14 (expressed by (a1(k) hat, a2(k) hat), determined by thecombination of the values of the identified gain coefficients a1(k) hat,a2(k) hat does not reside in the identifying coefficient limiting range,the value of a2(k) hat is forcibly changed to the lower limit value A2Lin STEP7-9-3. Thus, the point (a1(k) hat, a2(k) hat) on the coordinateplane shown in FIG. 14 is limited to a point in a region on and above astraight line (the straight line including the line segment Q₇Q₈)expressed by at least a2=A2L.

Then, the identifier 25 decides whether or not the value of theidentified gain coefficient a1(k) hat determined in STEP7-8 is equal toor greater than a lower limit value A1L for the gain coefficient a1 inthe identifying coefficient limiting range in STEP7-9-4, and thendecides whether or not the value of the identified gain coefficienta1(k) hat is equal to or smaller than an upper limit value A1H for thegain coefficient a1 in the identifying coefficient limiting range inSTEP7-9-5. In the present embodiment, the lower limit value A1L for thegain coefficient a1 is a predetermined fixed value. The upper limitvalue A1H for the gain coefficient a1 is represented by A1H=1−A2Lbecause it is an a1 coordinate of the point Q₈ where the polygonal line|a1|+a2=1 (a1>0) and the straight line a2=A2L intersect with each other,as shown in FIG. 14.

If a1(k) hat<A1L or a1(k) hat>A1H, then since the point (a1(k) hat,a2(k) hat) on the coordinate plane shown in FIG. 14 does not reside inthe identifying coefficient limiting range, the value of a1(k) hat isforcibly changed to the lower limit value A1L or the upper limit valueA1H in STEP7-9-5 and STEP7-9-7.

Thus, the point (a1(k) hat, a2(k) hat) on the coordinate plane shown inFIG. 14 is limited to a region on and between a straight line (thestraight line including the line segment Q₆A₇) expressed by a1=A1L, anda straight line (the straight line passing through the point Q₈ andperpendicular to the a1 axis) expressed by a1=A1H.

The processing in STEP7-9-4 through STEP7-9-7 may be carried out beforethe processing in STEP7-9-2 and STEP7-9-3.

Then, the identifier 25 decides whether the present values of a1(k) hat,a2(k) hat after STEP7-9-2 through STEP7-9-7 satisfy an inequality|a1|+a2≦1 or not, i.e., whether the point (a1(k) hat, a2(k) hat) ispositioned on or below or on or above the polygonal line (including linesegments Q₅Q₆ and Q₅Q₈) expressed by the functional expression |a1|+a2=1in STEP7-9-8.

If |a1|+a2=1, then the point (a1(k) hat, a2(k) hat) determined by thevalues of a1(k) hat, a2(k) hat after the processing in STEP7-9-2 throughSTEP7-9-7 exists in the identifying coefficient limiting range(including its boundaries).

If |a1|+a2>1, then since the point (a1(k) hat, a2(k) hat) deviatesupwardly from the identifying coefficient limiting range, the value ofthe a2(k) hat is forcibly changed to a value (1−|a1(k) hat|) dependingon the value of a1(k) hat in STEP7-9-9. Stated otherwise, while thevalue of a1(k) hat is being kept unchanged, the point (a1(k) hat, a2(k)hat) is moved onto a polygonal line expressed by the functionalexpression |a1|+a2=1 (onto the line segment Q₅Q₆ or the line segmentQ₅Q₈ which is a boundary of the identifying coefficient limiting range).

Through the above processing in STEP7-9-2 through 7-9-9, the values ofthe identified gain coefficients a1(k) hat, a2(k) hat are limited suchthat the point (a1(k) hat, a2(k) hat) determined thereby resides in theidentifying coefficient limiting range. If the point (a1(k) hat, a2(k)hat) corresponding to the values of the identified gain coefficientsa1(k) hat, a2(k) hat that have been determined in STEP7-8 exists in theidentifying coefficient limiting range, then those values of theidentified gain coefficients a1(k) hat, a2(k) hat are maintained.

The value of the identified gain coefficient a1(k) hat relative to theprimary autoregressive term of the discrete-system model is not forciblychanged insofar as the value resides between the lower limit value A1Land the upper limit value A1H of the identifying coefficient limitingrange. If a1(k) hat<A1L or a1(k) hat>A1H, then since the value of theidentified gain coefficient a1(k) hat is forcibly changed to the lowerlimit value A1L which is a minimum value that the gain coefficient a1can take in the identifying coefficient limiting range or the upperlimit value A1H which is a maximum value that the gain coefficient a1can take in the identifying coefficient limiting range, the change inthe value of the identified gain coefficient a1(k) hat is minimum.Stated otherwise, if the point (a1(k) hat, a2(k) hat) corresponding tothe values of the identified gain coefficients a1(k) hat, a2(k) hat thathave been determined in STEP7-8 deviates from the identifyingcoefficient limiting range, then the forced change in the value of theidentified gain coefficient a1(k) hat is held to a minimum.

After having limited the values of the identified gain coefficientsa1(k) hat, a2(k) hat, the identifier 25 performs a process of limitingthe value of the identified gain coefficient b1(k) hat in STEP7-9-10through STEP7-9-13.

Specifically, the identifier 25 decides whether or not the value of theidentified gain coefficient b1(k) hat determined in STEP7-8 is equal toor greater than the lower limit value B1L for the gain coefficient b1set in STEP7-9-1 in STEP7-9-10. If B1L>b1(k) hat, then the value ofb1(k) hat is forcibly changed to the lower limit value B1L inSTEP7-9-11.

The identifier 25 decides whether or not the value of the identifiedgain coefficient b1(k) hat is equal to or smaller than the upper limitvalue B1H for the gain coefficient g1 set in STEP7-9-1 in STEP7-9-12. IfB1H<b1(k) hat, then the value of b1(k) hat is forcibly changed to theupper limit value B1H in STEP7-9-13.

Through the above processing in STEP7-9-10 through 7-9-13, the value ofthe identified gain coefficient b1(k) hat is limited to a value in arange between the lower limit value B1L and the upper limit value B1H.

After the identifier 25 has limited the combination of the values of theidentified gain coefficients a1(k) hat, a2(k) hat and the identifiedgain coefficient b1(k) hat, control returns to the flowchart shown inFIG. 12.

The preceding values a1(k−1) hat, a2(k−1) hat, b1(k−1) hat of theidentified gain coefficients used for determining the identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat in STEP7-8 shown in FIG. 12are the values of the identified gain coefficients limited by thelimiting process in STEP7-9 in the preceding control cycle.

The above process is the processing sequence of the identifier 25 whichis carried out in STEP7 shown in FIG. 10.

In FIG. 10, after the processing of the identifier 25 has been carriedout, the exhaust-side control unit 7 a determines the values of the gaincoefficients a1, a2, b1 in STEP8. Specifically, if the value of the flagf/id/cal set in STEP2 is “1”, i.e., if the gain coefficients a1, a2, b1have been identified by the identifier 25, then the gain coefficientsa1, a2, b1 are set to the latest identified gain coefficients a1(k) hat,a2(k) hat, b1(k) hat determined by the identifier 25 in STEP7 (limitedin STEP7-9). If f/id/cal=“0”, i.e., if the gain coefficients a1, a2, b1have not been identified by the identifier 25, then the gaincoefficients a1, a2, b1 are set to predetermined values, respectively.

Then, the exhaust-side control unit 7 a effects a processing operationof the estimator 26 in STEP9.

The estimator 26 calculates the coefficients α1, α2, βj (j=1, 2, . . . ,d) to be used in the equation (14) or (15), using the gain coefficientsa1, a2, b1 determined in STEP8 (these values are basically the latestvalues of the identified gain coefficients a1 hat, a2 hat, b1 hat) andthe set dead time d1 of the exhaust system E and the set dead time d2 ofthe air-fuel ratio manipulating system, which have been set in STEP4,according to the definition with respect to the equation (13).

Then, the estimator 26 calculates the estimated differential outputVO2(k+d) bar (estimated value of the differential output VO2 after thetotal set dead time d from the time of the present control cycle)according to the equation (14), using the time-series data VO2(k),VO2(k−1) of the present and past values of the differential output VO2of the O₂ sensor calculated in each control cycle in STEP5, thetime-series data kact(k−j) (j=0, . . . , d1) of the present and pastvalues of the differential output kact of the LAF sensor 5, the datakcmd(k−j) (=Usl(k−j), j=1, . . . , d2−1) of the past values of thetarget differential air-fuel ratio kcmd (=the SLD manipulating inputUsl) given in each control cycle from the sliding mode controller 27,and the coefficients α1, α2, βj (j=1, 2, . . . , d) calculated asdescribed above.

Then, if the set dead time d2 of the air-fuel ratio manipulating systemis d2=1, then the estimator 26 calculates the estimated differentialoutput VO2(k+d) bar according to the equation (15), using thetime-series data VO2(k), VO_(2(k−)1) of the present and past values ofthe differential output VO2 of the O₂ sensor, time-series data kact(k−j)(j=0, . . . , d−1) of the present and past values of the differentialoutput kact of the LAF sensor 5, and the coefficients α1, α2, βj (j=1,2, . . . , d).

Then, the exhaust-side control unit 7 a calculates the SLD manipulatinginput Usl (=the target differential air-fuel ratio kcmd) with thesliding mode controller 27 in STEP10.

Specifically, the sliding mode controller 27 calculates a present valueσ(k+d) bar (corresponding to an estimated value, after the total setdead time d, of the linear function σ defined according to the equation(16)) of the switching function σ bar defined according to the equation(25), using the time-series data VO2(k+d) bar, VO2(k+d−1) bar (thepresent and preceding values of the estimated differential output VO2bar) of the estimated differential output VO2 bar determined by theestimator 26 in STEP9.

At this time, the sliding mode controller 27 keeps the value of theswitching function a bar within a predetermined allowable range. If thevalue σ(k+d) bar determined as described above exceeds the upper orlower limit of the allowable range, then the sliding mode controller 27forcibly limits the value σ(k+d) bar to the upper or lower limit of theallowable range.

Then, the sliding mode controller 27 accumulatively adds values σ(k+d)bar·ΔT, produced by multiplying the present value σ(k+d) bar of theswitching function σ bar by the period ΔT of the control cycles of theexhaust-side control unit 7 a. That is, the sliding mode controller 27adds the product σ(k+d) bar·ΔT of the value σ(k+d) bar and the period ΔTcalculated in the present control cycle to the sum determined in thepreceding control cycle, thus calculating an integrated value a bar(hereinafter represented by “Σσ bar”) which is the calculated result ofthe term Σ(σ bar·T) of the equation (27).

In the present embodiment, the sliding mode controller 27 keeps theintegrated value Σσ bar in a predetermined allowable range. If theintegrated value Σσ bar exceeds the upper or lower limit of theallowable range, then the sliding mode controller 27 forcibly limits theintegrated value Σσ bar to the upper or lower limit of the allowablerange.

Then, the sliding mode controller 27 calculates the equivalent controlinput Ueq, the reaching law input Urch, and the adaptive law input Uadpaccording to the respective equations (24), (26), (27), using thetime-series data VO2(k+d)bar, VO2(k+d−1) bar of the present and pastvalues of the estimated differential output VO2 bar determined by theestimator 26 in STEP9, the value σ(k+d) bar of the switching function aand its integrated value Σσ bar which are determined as described above,and the gain coefficients a1, a2, b1 determined in STEP 8 (which arebasically the latest identified gain coefficients a1(k) hat, a2(k) hat,b1(k) hat).

The sliding mode controller 27 then adds the equivalent control inputUeq, the reaching law input Urch, and the adaptive law input Uadp tocalculate the SLD manipulating input Usl, i.e., the input quantity (=thetarget differential air-fuel ratio kcmd) to be applied to the exhaustsystem E for converging the output signal VO2/OUT of the O₂ sensor 6 tothe target value VO2/TARGET.

After having calculated the SLD manipulating input Usl, the exhaust-sidecontrol unit 7 a determines the stability of the adaptive sliding modecontrol process (or more specifically, the stability of the controlledstate (hereinafter referred to as “SLD controlled state”) of the outputVO2/OUT of the O₂ sensor 6 based on the adaptive sliding mode controlprocess), and sets a value of a flag f/sld/stb indicative of whether theSLD controlled state is stable or not in STEP11. The flag f/sld/stb is“1” when the SLD controlled state is stable, and “0” when the SLDcontrolled state is not stable.

The stability determining process is carried out according to aflowchart shown in FIG. 17.

As shown in FIG. 17, the sliding mode controller 27 calculates adifference Δσ bar (corresponding to a rate of change of the switchingfunction σ bar) between the present value σ(k+d) bar of the switchingfunction σ bar calculated in STEP10 and a preceding value σ(k+d−1) barthereof in STEP11-1.

Then, the sliding mode controller 27 decides whether or not a product Δσbar·σ(k+d) bar (corresponding to the time-differentiated function of aLyapunov function a bar²/2 relative to the σ bar) of the difference Δσbar and the present value σ(k+d) bar of the switching function σ bar isequal to or smaller than a predetermined value ε (≧0) in STEP11-2.

The product Δσ bar·σ(k+d) bar (hereinafter referred to as “stabilitydetermining parameter Pstb”) will be described below. If the stabilitydetermining parameter Pstb is greater than 0 (Pstb>0), then the value ofthe switching function σ bar is basically shifting away from “0”. If thestability determining parameter Pstb is equal to or smaller than 0(Pstb≦0), then the value of the switching function σ bar is basicallyconverged or converging to “0”. Generally, in order to converge acontrolled variable to its target value according to the sliding modecontrol process, it is necessary that the value of the switchingfunction be stably converged to “0”. Basically, therefore, it ispossible to determine whether the SLD controlled state is stable orunstable depending on whether or not the value of the stabilitydetermining parameter Pstb is equal to or smaller than 0.

If, however, the stability of the SLD controlled state is determined bycomparing the value of the stability determining parameter Pstb with“0”, then the determined result of the stability is affected even byslight noise contained in the value of the switching function σ bar.According to the present embodiment, therefore, the predetermined valueε with which the stability determining parameter Pstb is to be comparedin STEP11-2 is of a positive value slightly greater than “0”.

If Pstb>ε in STEP11-2, then the SLD controlled state is judged as beingunstable, and the value of a timer counter tm (count-down timer) is setto a predetermined initial value T_(M) (the timer counter tm is started)in order to inhibit the determination of the target air-fuel ratio KCMDusing the SLD manipulating input Usl calculated in STEP10 for apredetermined time in STEP11-4. Thereafter, the value of the flagf/sld/stb is set to “0” in STEP11-5, after which control returns to themain routine shown in FIG. 10.

If Pstb≦ε in STEP11-2, then the sliding mode controller 27 decideswhether the present value σ(k+d) bar of the switching function σ barfalls within a predetermined range or not in STEP11-3.

If the present value σ(k+d) bar of the switching function σ bar does notfall within the predetermined range, then since the present value σ(k+d)bar is spaced widely apart from “0”, the SLD controlled state isconsidered to be unstable. Therefore, if the present value σ(k+d) bar ofthe switching function σ bar does not fall within the predeterminedrange in STEP11-3, then the SLD controlled state is judged as beingunstable, and the processing of STEP11-4 and STEP11-5 is executed tostart the timer counter tm and set the value of the flag f/sld/stb to“0”.

In the present embodiment, since the value of the switching function σbar is limited within the allowable range in STEP10, the decisionprocessing in STEP11-3 may be dispensed with.

If the present value σ(k+d) bar of the switching function σ bar fallswithin the predetermined range in STEP11-3, then the sliding modecontroller 27 counts down the timer counter tm for a predetermined timeΔtm in STEP11-6. The sliding mode controller 27 then decides whether ornot the value of the timer counter tm is equal to or smaller than “0”,i.e., whether a time corresponding to the initial value T_(M) haselapsed from the start of the timer counter tm or not, in STEP11-7.

If tm>0, i.e., if the timer counter tm is still measuring time and itsset time has not yet elapsed, then since no substantial time has elapsedafter the SLD controlled state is judged as unstable in STEP11-2 orSTEP11-3, the SLD controlled state tends to become unstable. Therefore,if tm>0 in STEP11-7, then the value of the flag f/sld/stb is set to “0”in STEP11-5.

If tm≦0 in STEP11-7, i.e., if the set time of the timer counter tm haselapsed, then the SLD controlled stage is judged as being stable, andthe value of the flag f/sld/stb is set to “1” in STEP11-8.

According to the above processing, the stability of the SLD controlledstate is determined. If the SLD controlled state is judged as beingunstable, then the value of the flag f/sld/stb is set to “0”, and if theSLD controlled state is judged as being stable, then the value of theflag f/sld/stb is set to “1”.

The above process of determining the stability of the SLD controlledstate is by way of illustrative example only. The stability of the SLDcontrolled state may be determined by any of various other processes.For example, in each given period longer than the control cycle, thefrequency with which the value of the stability determining parameterPstb in the period is greater than the predetermined value ε is counted.If the frequency is in excess of a predetermined value, then the SLDcontrolled state is judged as unstable. Otherwise, the SLD controlledstate is judged as stable.

Referring back to FIG. 10, after a value of the flag f/sld/stbindicative of the stability of the SLD controlled state has been set,the sliding mode controller 27 determines the value of the flagf/sld/stb in STEP12. If the value of the flag f/sld/stb is “1”, i.e., ifthe SLD controlled state is judged as being stable, then the slidingmode controller 27 limits the SLD manipulating input Usl calculated inSTEP10 in STEP13. Specifically, the sliding mode controller 27determines whether the present value Usl(k) of the SLD manipulatinginput Usl calculated in STEP10 falls in a predetermined allowable rangeor not. If the present value Usl exceeds the upper or lower limit of theallowable range, then the sliding mode controller 27 forcibly limits thepresent value Usl(k) of the SLD manipulating input Usl to the upper orlower limit of the allowable range.

The SLD manipulating input Usl (=the target differential air-fuel ratiokcmd) limited in STEP13 is stored in a memory (not shown) in atime-series fashion, and will be used in the processing operation of theestimator 26.

Then, the sliding mode controller 27 adds the air-fuel ratio referencevalue FLAF/BASE to the SLD manipulating input Usl limited in STEP13,thus calculating the target air-fuel ratio KCMD in STEP15. Theprocessing in the present control cycle of the exhaust-side control unit7 a is now put to an end.

If f/sld/stb=0 in STEP12, i.e., if the SLD controlled state is judged asunstable, then the sliding mode controller 27 forcibly sets the value ofthe SLD manipulating input Usl in the present control cycle to apredetermined value (the fixed value or the preceding value of the SLDmanipulating input Usl) in STEP14. The sliding mode controller 27calculates the target air-fuel ratio KCMD by adding the air-fuel ratioreference value FLAF/BASE to the SLD manipulating input Usl in STEP15.The processing in the present control cycle of the exhaust-side controlunit 7 a is now put to an end.

The target air-fuel ratio KCMD finally determined in STEP15 is stored ina memory (not shown) in a time-series fashion in each control cycle.When the general feedback controller 15 is to use the target air-fuelratio KCMD determined by the exhaust-side control unit 7 a (see STEPf inFIG. 8), the latest one of the time-series data of the target air-fuelratio KCMD thus stored is selected.

Details of the operation of the apparatus according to the presentembodiment have been described above.

The operation of the apparatus will be summarized as follows: Theexhaust-side control unit 7 a sequentially determines the targetair-fuel ratio KCMD which is a target value for the upstream-of-catalystair-fuel ratio so as to converge (adjust) the output signal VO2/OUT ofthe O₂ sensor 6 downstream of the catalytic converter 3 to the targetvalue VO2/TARGET therefor. The amount of fuel injected into the internalcombustion engine 1 is adjusted to converge the output of the LAF sensor5 to the target air-fuel ratio KCMD, thereby feedback-controlling theupstream-of-catalyst air-fuel ratio at the target air-fuel ratio KCMD,and hence converging the output VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET. The catalytic converter 3 can thus maintain itsoptimum exhaust gas purifying performance.

In this case, in order to calculate the target air-fuel ratio KCMDaccording to the adaptive sliding mode control process of the slidingmode controller 27, the exhaust-side control unit 7 a uses the estimateddifferential output VO2 bar determined by the estimator 27, i.e., theestimated differential output VO2 bar which is an estimated value of thedifferential output VO2 of the O₂ sensor 6 after the total set dead timed which is the sum of the set dead time d1 of the exhaust system E andthe set dead time d2 of the air-fuel ratio manipulating system (thesystem comprising the internal combustion engine 1 and the engine-sidecontrol unit 7 b). The exhaust-side control unit 7 a determines thetarget air-fuel ratio KCMD so as to converge the estimated value of theoutput VO2/OUT of the O₂ sensor 6 after the total set dead time d whichis represented by the estimated differential output VO2 bar.

The estimated differential output VO2 bar determined by the estimator 26is the estimated value of the differential output VO2 of the O₂ sensor 6after the set dead times d1, d2 set by the dead time setting means 29depending on the estimated exhaust gas volume ABSV determined by theflow rate data generating means 28, i.e., the total set dead time ddetermined by the set dead times d1, d2 that are substantially equal tothe actual dead times of the exhaust system E and the air-fuel ratiomanipulating system. The algorithm for calculating the estimateddifferential output VO2 bar with the estimator 26 is constructed on thebasis of the exhaust system model and the air-fuel ratio manipulatingsystem model which have the respective dead time elements of the setdead times d1, d2. The values of the gain coefficients a1, a2, b1 whichare parameters of the exhaust system model are sequentially identifiedto minimize an error between the identified differential output VO2 hatindicative of the differential output VO2 of the O₂ sensor 6 on theexhaust system model and the actual differential output VO2, and theidentified values a1 hat, a2 hat, b1 hat thereof are used in the processof calculating the estimated differential output VO2 bar with theestimator 26. Since the set dead time d1 that is substantially equal tothe actual dead time of the exhaust system E is used as the dead time ofthe exhaust system model, the matching between the exhaust system modeland the behavioral characteristics of the actual exhaust system E isincreased, allowing the identifier 25 to determine the identified gaincoefficients a1 hat, a2 hat, b1 hat which accurately reflect the actualbehavior of the exhaust system E.

The estimated differential output VO2 bar determined by the estimator 26is thus highly accurate, not depending on changes in the actual deadtimes of the exhaust system E and the air-fuel ratio manipulatingsystem, but representing the output of the O₂ sensor 6 after the totaldead time which is the sum of those dead times. Using the estimateddifferential output VO2 bar, the sliding mode controller 27 candetermine the target air-fuel ratio KCMD which is capable of optimallycompensating for the effect of the dead times of the exhaust system Eand the air-fuel ratio manipulating system, and hence can perform thecontrol process of converging the output VO2/OUT of the O₂ sensor 6 tothe target value VO2/TARGET accurately with a highly quick response. Asa result, the purifying capability of the catalytic converter 3 can beincreased.

The algorithm of the adaptive sliding mode control process of thesliding mode controller 27 for determining the target air-fuel ratioKCMD is constructed on the basis of the exhaust system model having theset dead time d1 which is substantially equal to the actual dead time ofthe exhaust system E, as with the estimator 26, and uses the identifiedgain coefficients a1 hat, a2 hat, b1h hat that are sequentiallydetermined by the identifier 25 in order to determine the targetair-fuel ratio KCMD. Therefore, the target air-fuel ratio KCMD can bedetermined to as to accurately reflect the actual behavior of theexhaust system E, and the quick response of the control process ofconverging the output VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET can be increased to increase the purifying capability of thecatalytic converter 3.

The identifier 25 limits combinations of the identified gaincoefficients a1 hat, a2 hat to be determined to values within theidentifying coefficient limiting range that is variably establisheddepending on the estimated exhaust gas volume ABSV which determines theset dead times d1, d2, and also sets the value of the identified gaincoefficient b1 to a value within the range that is also variablyestablished depending on the estimated exhaust gas volume ABSV. Theidentifier 25 variably adjusts the value of the weighted parameter λ₁ inthe algorithm of the method of weighted least squares for determiningthe identified gain coefficients a1 hat, a2 hat, b1 hat, depending onthe estimated exhaust gas volume ABSV. Therefore, errors and variationsof these identified gain coefficients a1 hat, a2 hat, b1 hat can besuppressed and their reliability is increased, without depending onchanges in the actual dead times and the response delay characteristicsof the exhaust system E and the air-fuel ratio manipulating system. As aresult, the accuracy of the estimated differential output VO2 that isdetermined by the estimator 26 using the identified gain coefficients a1hat, a2 hat, b1 hat can stably be maintained, and the target air-fuelratio KCMD that is capable of converging the output VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET smoothly with a highly quickresponse can stably be determined. Thus, the high purifying capabilityof the catalytic converter 3 can stably be maintained.

A second embodiment of the present invention will be described below.The present embodiment is an embodiment relating to the first aspect ofthe present invention. The present embodiment basically differs from theprevious embodiment as to only the processing operation of the estimator26, and employs the same reference characters as those of the previousembodiment for its description.

In the previous embodiment, the estimated value of the differentialoutput VO2 of the O₂ sensor 6 after the total set dead time d (=d1+d2)is determined in order to compensate for the effect of both the deadtime d1 of the exhaust system E and the dead time d2 of the air-fuelratio manipulating system (the system comprising the internal combustionengine 1 and the engine-side control unit 7 b). However, if the deadtime d2 of the air-fuel ratio manipulating system is sufficiently small(it can be regarded as d2≈0) compared with the dead time d1 of theexhaust system E, then an estimated value VO2(k+d1) bar (hereinafterreferred to as “second estimated differential output VO2(k+d1) bar”) ofthe differential output VO2 of the O₂ sensor 6 after the dead time d1 ofthe exhaust system E may be determined, and the target air-fuel ratioKCMD may be determined using the second estimated differential outputVO2(k+d1) bar. According to the present embodiment, the second estimateddifferential output VO2(k+d1) bar is determined, and the output VO2/OUTof the O₂ sensor 6 is converged to the target value VO2/TARGET.

The estimator 26 determines the second estimated differential outputVO2(k+d1) bar as follows: Using the equation (1) expressing the exhaustsystem model of the exhaust system E, the second estimated differentialoutput VO2(k+d1) bar which is an estimated value VO2(k+d1) bar of thedifferential output VO2 of the O₂ sensor 6 after the dead time d1 of theexhaust system E in each control cycle is expressed by the followingequation (42), using the time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 and the time-series datakact(k−j) (j=1, 2, . . . , d1) of the past values of the differentialoutput kact (=KACT−FLAF/BASE) of the LAF sensor 5: $\begin{matrix}{{\overset{\_}{VO2}\left( {k + {d1}} \right)} = {{{\alpha 3} \cdot {{VO2}(k)}} + {{\alpha 4} \cdot {{VO2}\left( {k - 1} \right)}} + {\sum\limits_{j = 1}^{d1}{\gamma_{j} \cdot {{kact}\left( {k - j} \right)}}}}} & (42)\end{matrix}$where

-   -   α3=the first-row, first-column element of A^(d1),    -   α4=the first-row, second-column element of A^(d1),    -   γj=the first-row elements of A^(j−1)·B $A = \begin{bmatrix}        {a1} & {a2} \\        1 & 0        \end{bmatrix}$ $B = \begin{bmatrix}        {b1} \\        0        \end{bmatrix}$

In the equation (42), “α3”, “α4” represent the first-row, first-columnelement and the first-row, second-column element, respectively, of thepower A_(d1) (d1: dead time of the exhaust system E) of the matrix Adefined as described above with respect to the equation (13), and “γj”(j=1, 2, . . . , d1) represents the first-row elements of the productA^(j−1)·B of the power A^(j−1) (j=1, 2, . . . , d1) of the matrix A andthe vector B defined as described above with respect to the equation(13).

The equation (42) is an equation for the estimator 26 to calculate thesecond estimated differential output VO2(k+d1) bar. The equation (42) isobtained from the equation (13) by setting kcmd(k)=kact(k), d=d1 (thedead time d2 of the air-fuel ratio manipulating system is regarded as“0”) in the equation (18) described in the first embodiment. In thepresent embodiment, therefore, the estimator 26 determines, in eachcontrol cycle, calculates the equation (42) to determine the secondestimated differential output VO2(k+d1) bar of the O₂ sensor 6, usingthe time-series data VO2(k), VO2(k−1) of the differential output VO2 ofthe O₂ sensor 6 and the time-series data kact(k−j) (j=1, 2, . . . , d1)of the past values of the differential output kact of the LAF sensor 5.

The values of the coefficients α3, α4, γj (j=1, 2, . . . , d1) requiredto calculate the second estimated differential output VO2(k+d1) baraccording to the equation (42) are calculated using the identified gaincoefficients a1 hat, a2 hat, b1 hat which represent the identifiedvalues of the gain coefficients a1, a2, b1. The value of the dead timed1 required in the calculation of the equation (42) employs the set deadtime d1 that is sequentially determined in each control cycle by thedead time setting means 29, as with the first embodiment. In this case,the dead time setting means 29 is not required to determine the set deadtime d2 of the air-fuel ratio manipulating system.

Other processing details than described above are basically the same asthose of the first embodiment. However, the sliding mode controller 27determines the equivalent control input Ueq, the reaching law inputUrch, and the adaptive law input Uadp, which are components of the SLDmanipulating input Usl, according to the equations (24), (26), (27)where “d” is replaced with “d1”.

With the apparatus for controlling the air-fuel ratio of the internalcombustion engine according to the present embodiment, the set dead timed1 of the exhaust system E to be taken into account in converging theoutput VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET isvariably set depending on the estimated exhaust gas volume so as to besubstantially equal to the actual dead time. Using the value of the setdead time d1, the processing sequences of the identifier 25, theestimator 26, and the sliding mode controller 27 are carried out in thesame manner as with the first embodiment. Therefore, the presentembodiment offers the same advantages as those of the first embodiment.

The apparatus for controlling the air-fuel ratio according to thepresent invention is not limited to the above embodiments, but may bemodified as follows:

In the first and second embodiments, the O₂ sensor 6 is used as theexhaust gas sensor downstream of the catalytic converter 3. However, anyof various other sensors may be employed insofar as they can detect theconcentration of a certain component of the exhaust gas downstream ofthe catalytic converter to be controlled. For example, a CO sensor isemployed if the carbon monoxide (CO) in the exhaust gas downstream ofthe catalytic converter is controlled, an NOx sensor is employed if thenitrogen oxide (NOx) in the exhaust gas downstream of the catalyticconverter is controlled, and an HC sensor is employed if the hydrocarbon(HC) in the exhaust gas down-stream of the catalytic converter iscontrolled.

In the above embodiments, the differential output kact of the LAF sensor5, the differential output VO2 of the O₂ sensor 6, and the targetdifferential air-fuel ratio kcmd are employed in the processingsequences of the identifier 25, the estimator 26, and the sliding modecontroller 27. However, the processing sequences of the identifier 25,the estimator 26, and the sliding mode controller 27 may be performeddirectly using the output KACT of the LAF sensor 5, the output VO2/OUTof the O₂ sensor 6, and the target air-fuel ratio KCMD.

In the above embodiments, the manipulated variable generated by theexhaust-side control unit 7 a is the target air-fuel ratio KCMD (thetarget input for the exhaust system E), and the air-fuel ratio of theair-fuel mixture to be combusted by the internal combustion engine 1 ismanipulated according to the target air-fuel ratio KCMD. However, acorrected amount of the amount of fuel supplied to the internalcombustion engine 1 may be determined by the exhaust-side control unit 7a, and the amount of fuel supplied to the internal combustion engine 1may be adjusted in a feed-forward fashion from the target air-fuel ratioKCMD to manipulate the air-fuel ratio.

In the above embodiments, the sliding mode controller 27 employs anadaptive sliding mode control process which incorporates an adaptive law(adaptive algorithm) taking into account the effect of disturbances.However, the sliding mode controller 27 may employ a normal sliding modecontrol process which is free from such an adaptive law. Furthermore,the sliding mode controller 27 may be replaced with another type ofadaptive controller, e.g., a back-stepping controller or the like.

Industrial Applicability

As described above, the present invention is useful for controlling theair-fuel ratio of an internal combustion engine mounted on an automobileor the like to increase the exhaust gas purifying capability of acatalytic converter.

1. An apparatus for controlling the air-fuel ratio of an internalcombustion engine having an exhaust gas sensor disposed downstream of acatalytic converter disposed in an exhaust passage of the internalcombustion engine, for detecting the concentration of a particularcomponent in an exhaust gas which has passed through the catalyticconverter, estimating means for sequentially generating datarepresentative of an estimated value of an output of said exhaust gassensor after a set dead time, using the set dead time which is set as adead time of an exhaust system ranging from a position upstream of saidcatalytic converter to said exhaust gas sensor and including saidcatalytic converter, manipulated variable generating means forgenerating a manipulated variable to determine an air-fuel ratio of theexhaust gas which enters said catalytic converter to converge the outputof said exhaust gas sensor to a predetermined target value, using thedata generated by said estimating means, and air-fuel ratio manipulatingmeans for manipulating the air-fuel ratio of an air-fuel mixture to becombusted by the internal combustion engine depending on the manipulatedvariable, comprising: flow rate data generating means for sequentiallygenerating data representative of a flow rate of the exhaust gassupplied to the catalytic converter, and dead time setting means forvariably setting a value of said set dead time depending on the value ofthe data generated by said flow rate data generating means; wherein apredetermined model of said exhaust system is established for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of said exhaust gas sensor via a dead time elementand a response delay element of said set dead time from the air-fuelratio of the exhaust gas which enters said catalytic converter, furthercomprising identifying means for sequentially identifying the value of apredetermined parameter of said model using the value of the set deadtime set by said dead time setting means; wherein said estimating meansgenerates the data representative of the estimated value of the outputof said exhaust gas sensor using the identified value of said parameterdetermined by said identifying means, according to a predeterminedestimating algorithm which is constructed based on the model of saidexhaust system; and wherein said identifying means determines theidentified value of the parameter of the model of said exhaust systemfor limiting the identified value to a value within a predeterminedrange depending on the value of the data generated by said flow ratedata generating means.
 2. An apparatus for controlling the air-fuelratio of an internal combustion engine according to claim 1, whereinsaid identifying means comprises means for identifying the value of saidparameter according to an algorithm for minimizing an error between theoutput of said exhaust gas sensor in the model of said exhaust systemand an actual output of said exhaust gas sensor, further comprisingmeans for variably setting the value of a weighted parameter of saidalgorithm depending on the value of the data generated by said flow ratedata generating means.
 3. An apparatus for controlling the air-fuelratio of an internal combustion engine according to claim 1, whereinsaid manipulated variable generating means generates said manipulatedvariable using the identified value, determined by said identifyingmeans, of the parameter of said model of said exhaust system.
 4. Arecording medium readable by a computer and storing an air-fuel ratiocontrol program for enabling said computer to perform a process ofsequentially generating data representative of an estimated value of anoutput of an exhaust gas sensor disposed downstream of a catalyticconverter disposed in an exhaust passage of the internal combustionengine, for detecting the concentration of a particular component in anexhaust gas which has passed through the catalytic converter, after aset dead time which is set as a dead time of an exhaust system rangingfrom a position upstream of said catalytic converter to said exhaust gassensor and including said catalytic converter, a process of generating amanipulated variable to determine an air-fuel ratio of the exhaust gaswhich enters said catalytic converter to converge the output of saidexhaust gas sensor to a predetermined target value, using the datarepresentative of the estimated value, and a process of manipulating theair-fuel ratio of an air-fuel mixture to be combusted by the internalcombustion engine depending on the manipulated variable, said air-fuelratio control program comprising: a program of enabling the computer toperform a process of sequentially generate data representative of a flowrate of the exhaust gas supplied to the catalytic converter, andvariably setting a value of said set dead time depending on the value ofthe data representative of the flow rate of the exhaust gas wherein apredetermined model of said exhaust system is established for expressinga behavior of the exhaust system which is regarded as a system forgenerating the output of said exhaust gas sensor via a dead time elementand a response delay element of said set dead time from the air-fuelratio of the exhaust gas which interes said catalytic converter, saidair-fuel ratio control program includes a program for enabling thecomputer to perform a process of sequentially identifying the value of apredetermined parameter of said model using the value of the set deadtime set by said dead time setting means; wherein the program of saidair-fuel ratio control program for generating the data representative ofthe estimated value of the output of the exhaust gas sensor enables thecomputer to generate the data representative of the estimated value ofthe output of said exhaust gas sensor using the identified value of saidparameter, according to a predetermined estimating algorithm which isconstructed based on the model of said exhaust system; and wherein theprogram of said air-fuel ratio control program for identifying theparameter of the model of said exhaust system determines the identifiedvalue of the parameter of the model of said exhaust system by limitingthe identified value to a value within a predetermined range dependingon the value of the data representative of the flow rate of the exhaustgas supplied to said catalytic converter.
 5. A recording medium storingan air-fuel ratio control program for an internal combustion engineaccording to claim 4, wherein the program of said air-fuel ratio controlprogram for identifying the parameter of the model of said exhaustsystem identifies the value of said parameter according to an algorithmfor minimizing an error between the output of said exhaust gas sensor inthe model of said exhaust system and an actual output of said exhaustgas sensor, and variably sets the value of a weighted parameter of saidalgorithm depending on the value of the data representing the flow rateof said exhaust gas.
 6. A recording medium storing an air-fuel ratiocontrol program for an internal combustion engine according to claim 4,wherein the program of said air-fuel ratio control program forgenerating said manipulated variable is constructed of an algorithm forusing the identified value of the parameter of the model of said exhaustsystem in order to generate the manipulated variable.
 7. A method ofcontrolling the air-fuel ratio of an internal combustion engine,comprising the steps of sequentially generating data representative ofan estimated value of an output of an exhaust gas sensor disposeddownstream of a catalytic converter disposed in an exhaust passage ofthe internal combustion engine, for detecting the concentration of aparticular component in an exhaust gas which has passed through thecatalytic converter, after a set dead time which is set as a dead timeof an exhaust system ranging from a position upstream of said catalyticconverter to said exhaust gas sensor and including said catalyticconverter, and generating a manipulated variable to determine anair-fuel ratio of the exhaust gas which enters said catalytic converterto converge the output of said exhaust gas sensor to a predeterminedtarget value, using the data representative of the estimated value,wherein the air-fuel ratio of an air-fuel mixture to be combusted by theinternal combustion engine is manipulated depending on the manipulatedvariable, comprising the steps of: sequentially generating datarepresentative of a flow rate of the exhaust gas supplied to thecatalytic converter, and variably setting a value of said set dead timedepending on the value of the data representative of the flow rate ofthe exhaust gas; wherein a predetermined model of said exhaust system isestablished for expressing a behavior of the exhaust system which isregarded as a system for generating the output of said exhaust gassensor via a dead time element and a response delay element of said setdead time from the air-fuel ratio of the exhaust gas which enters saidcatalytic converter, further comprising the step of sequentiallyidentifying the value of a predetermined parameter of said model usingthe value of said set dead time; wherein said step of generating datarepresentative of the estimated value of the output of the exhaust gassensor generates the data representative of the estimated value of theoutput of said exhaust gas sensor using the identified value of saidparameter, according to a predetermined estimating algorithm which isconstructed based on the model of said exhaust system; and wherein saidstep of identifying the parameter of the model of said exhaust systemdetermines the identified value of the parameter of the model of saidexhaust system by limiting the identified value to a value within apredetermined range depending on the value of the data representative ofthe flow rate of the exhaust gas supplied to said catalytic converter.8. A method of controlling the air-fuel ratio of an internal combustionengine according to claim 7, wherein said step of identifying theparameter of the model of said exhaust system identifies the value ofsaid parameter according to an algorithm for minimizing an error betweenthe output of said exhaust gas sensor in the model of said exhaustsystem and an actual output of said exhaust gas sensor, and variablysets the value of a weighted parameter of said algorithm depending onthe value of the data representative of the flow rate of said exhaustgas.
 9. A method of controlling the air-fuel ratio of an internalcombustion engine according to claim 7, wherein said step of generatingsaid manipulated variable uses the identified value of the parameter ofthe model of said exhaust system determined by said identifying means inorder to generate said manipulated variable.
 10. An apparatus forcontrolling the air-fuel ratio of an internal combustion engine havingan exhaust gas sensor disposed downstream of a catalytic converterdisposed in an exhaust passage of the internal combustion engine, fordetecting the concentration of a particular component in an exhaust gaswhich has passed through the catalytic converter, manipulated variablegenerating means for sequentially generating a manipulated variable todetermine an air-fuel ratio of the exhaust gas which enters saidcatalytic converter to converge an output of said exhaust gas sensor toa predetermined target value, air-fuel ratio manipulating means formanipulating the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine depending on the manipulated variable,and estimating means for sequentially generating data representative ofan estimated value of the output of said exhaust gas sensor after atotal set dead time which is the sum of a first set dead time and asecond set dead time, said first set dead time being set as a dead timeof an exhaust system ranging from a position upstream of said catalyticconverter to said exhaust gas sensor and including said catalyticconverter, said second set dead time being set as a dead time of anair-fuel ratio manipulating system comprising said air-fuel ratiomanipulating means and said internal combustion engine, wherein saidmanipulated variable generating means generates said manipulatedvariable using the data generated by said estimating means, comprising:flow rate data generating means for sequentially generating datarepresentative of a flow rate of the exhaust gas supplied to thecatalytic converter, and dead time setting means for variably settingvalues of said first set dead time and said second set dead timedepending on the value of the data generated by said flow rate datagenerating means; wherein a predetermined model of said exhaust systemis established for expressing a behavior of the exhaust system which isregarded as a system for generating the output of said exhaust gassensor via a dead time element and a response delay element of saidfirst set dead time from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, further comprising identifying meansfor sequentially identifying the value of a predetermined parameter ofsaid model using the value of the first set dead time set by said deadtime setting means; wherein said estimating means generates theestimated value of the output of said exhaust gas sensor using theidentified value of said parameter determined by said identifying means,according to a predetermined estimating algorithm which is constructedbased on the model of said exhaust system and a predetermined model ofsaid air-fuel ratio manipulating means for expressing a behavior of theair-fuel ratio manipulating means which is regarded as a system forgenerating the air-fuel ratio detected by said air-fuel ratio sensorfrom said manipulated variable via a dead time element of said secondset dead time; and wherein said identifying means determines theidentified value of the parameter of the model of said exhaust system bylimiting the identified value to a value within a predetermined rangedepending on the value of the data generated by said flow rate datagenerating means.
 11. An apparatus for controlling the air-fuel ratio ofan internal combustion engine according to claim 10, wherein saididentifying means comprises means for identifying the value of saidparameter according to an algorithm for minimizing an error between theoutput of said exhaust gas sensor in the model of said exhaust systemand an actual output of said exhaust gas sensor, further comprisingmeans for variably setting the value of a weighted parameter of saidalgorithm depending on the value of the data generated by said flow ratedata generating means.
 12. An apparatus for controlling the air-fuelratio of an internal combustion engine according to claim 10, whereinsaid manipulated variable generating means generates said manipulatedvariable using the identified value, determined by said identifyingmeans, of the parameter of said model of said exhaust system.
 13. Arecording medium readable by a computer and storing an air-fuel ratiocontrol program for enabling said computer to perform a process ofsequentially generating a manipulated variable to determine an air-fuelratio of the exhaust gas which enters said catalytic converter toconverge an output of an exhaust gas sensor, which is disposeddownstream of a catalytic converter disposed in an exhaust passage ofthe internal combustion engine, for detecting the concentration of aparticular component in an exhaust gas which has passed through thecatalytic converter, to a predetermined target value, a process ofmanipulating the air-fuel ratio of an air-fuel mixture to be combustedby the internal combustion engine depending on the manipulated variable,and a process of sequentially generating data representative of anestimated value of the output of said exhaust gas sensor after a totalset dead time which is the sum of a first set dead time and a second setdead time, said first set dead time being set as a dead time of anexhaust system ranging from a position upstream of said catalyticconverter to said exhaust gas sensor and including said catalyticconverter, said second set dead time being set as a dead time of anair-fuel ratio manipulating system comprising said air-fuel ratiomanipulating means and said internal combustion engine, wherein theprogram of said air-fuel ratio control program for generating themanipulated variable is constructed of an algorithm for generating themanipulated variable using the data representative of the estimatedvalue of the output of the exhaust gas sensor, said air-fuel ratiocontrol program comprising: a program for enabling the computer toperform a process of sequentially generating data representative of aflow rate of the exhaust gas supplied to the catalytic converter, andvariably setting values of said first set dead time and said second setdead time depending on the value of the data representative of the flowrate of the exhaust gas; wherein a predetermined model of said exhaustsystem is established for expressing a behavior of the exhaust systemwhich is regarded as a system for generating the output of said exhaustgas sensor via a dead time element and a response delay element of saidfirst set dead time from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, said air-fuel ratio control programincludes a program for enabling the computer to perform a process ofsequentially identifying the value of a predetermined parameter of saidmodel using the value of said first set dead time; wherein the programof said air-fuel ratio control program for generating the datarepresentative of the estimated value of the output of said exhaust gassensor enables the computer to generate the estimated value of theoutput of said exhaust gas sensor using the identified value of saidparameter of the model of said exhaust system, according to apredetermined estimating algorithm which is constructed based on themodel of said exhaust system and a predetermined model of said air-fuelratio manipulating means for expressing a behavior of the air-fuel ratiomanipulating means which is regarded as a system for generating theair-fuel ratio detected by said air-fuel ratio sensor from saidmanipulated variable via a dead time element of said second set deadtime; and wherein the program of said air-fuel ratio control program foridentifying the parameter of the model of said exhaust system determinesthe identified value of the parameter of the model of said exhaustsystem by limiting the identified value to a value within apredetermined range depending on the value of the data representative ofthe flow rate of the exhaust gas supplied to said catalytic converter.14. A recording medium storing an air-fuel ratio control program for aninternal combustion engine according to claim 13, wherein the program ofsaid air-fuel ratio control program for identifying the parameter of themodel of said exhaust system identifies the value of said parameteraccording to an algorithm for minimizing an error between the output ofsaid exhaust gas sensor in the model of said exhaust system and anactual output of said exhaust gas sensor, and variably sets the value ofa weighted parameter of said algorithm depending on the value of thedata representing the flow rate of said exhaust gas.
 15. A recordingmedium storing an air-fuel ratio control program for an internalcombustion engine according to claim 13, wherein the program of saidair-fuel ratio control program for generating said manipulated variableis constructed of an algorithm for using the identified value of theparameter of the model of said exhaust system in order to generate themanipulated variable.
 16. A method of controlling the air-fuel ratio ofan internal combustion engine, comprising the steps of sequentiallygenerating a manipulated variable to determine an air-fuel ratio of theexhaust gas which enters said catalytic converter to converge an outputof an exhaust gas sensor, which is disposed downstream of a catalyticconverter disposed in an exhaust passage of the internal combustionengine, for detecting the concentration of a particular component in anexhaust gas which has passed through the catalytic converter, to apredetermined target value, manipulating the air-fuel ratio of anair-fuel mixture to be combusted by the internal combustion enginedepending on the manipulated variable, and sequentially generating datarepresentative of an estimated value of the output of said exhaust gassensor after a total set dead time which is the sum of a first set deadtime and a second set dead time, said first set dead time being set as adead time of an exhaust system ranging from a position upstream of saidcatalytic converter to said exhaust gas sensor and including saidcatalytic converter, said second set dead time being set as a dead timeof an air-fuel ratio manipulating system comprising said air-fuel ratiomanipulating means and said internal combustion engine, wherein saidstep of generating said manipulated variable uses the datarepresentative of the estimated value of the output of the exhaust gassensor in order to generate said manipulated variable, comprising thesteps of: sequentially generating data representative of a flow rate ofthe exhaust gas supplied to the catalytic converter, and variablysetting values of said first set dead time and said second set dead timedepending on the value of the data representative of the flow rate ofthe exhaust gas; wherein there is established a predetermined model ofsaid exhaust system for expressing a behavior of the exhaust systemwhich is regarded as a system for generating the output of said exhaustgas sensor via a dead time element and a response delay element of saidfirst set dead time from the air-fuel ratio of the exhaust gas whichenters said catalytic converter, further comprising the step ofsequentially identifying the value of a predetermined parameter of saidmodel using the value of said first set dead time; and wherein said stepof generating the data representative of the estimated value of theoutput of said exhaust gas sensor generates the estimated value of theoutput of said exhaust gas sensor using the identified value of saidparameter of the model of said exhaust system, according to apredetermined estimating algorithm which is constructed based on themodel of said exhaust system and a predetermined model of said air-fuelratio manipulating means for expressing a behavior of the air-fuel ratiomanipulating means which is regarded as a system for generating theair-fuel ratio detected by said air-fuel ratio sensor from saidmanipulated variable via a dead time element of said second set deadtime, and said step of identifying the parameter of the model of saidexhaust system determines the identified value of the parameter of themodel of said exhaust system by limiting the identified value to a valuewithin a predetermined range depending on the value of the datarepresentative of the flow rate of the exhaust gas supplied to saidcatalytic converter.
 17. A method of controlling the air-fuel ratio ofan internal combustion engine according to claim 16, wherein said stepof identifying the parameter of the model of said exhaust systemidentifies the value of said parameter according to an algorithm forminimizing an error between the output of said exhaust gas sensor in themodel of said exhaust system and an actual output of said exhaust gassensor, and variably sets the value of a weighted parameter of saidalgorithm depending on the value of the data representative of the flowrate of said exhaust gas.
 18. A method of controlling the air-fuel ratioof an internal combustion engine according to claim 16, wherein saidstep of generating said manipulated variable uses the identified valueof the parameter of the model of said exhaust system determined by saididentifying means in order to generate said manipulated variable.